%Version 3.1 December 2024
% See section 11 of the User Manual for version history
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%                                                                 %%
%% Please do not use \input{...} to include other tex files.       %%
%% Submit your LaTeX manuscript as one .tex document.              %%
%%                                                                 %%
%% All additional figures and files should be attached             %%
%% separately and not embedded in the \TeX\ document itself.       %%
%%                                                                 %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%\documentclass[referee,sn-basic]{sn-jnl}% referee option is meant for double line spacing

%%=======================================================%%
%% to print line numbers in the margin use lineno option %%
%%=======================================================%%

%%\documentclass[lineno,pdflatex,sn-basic]{sn-jnl}% Basic Springer Nature Reference Style/Chemistry Reference Style

%%=========================================================================================%%
%% the documentclass is set to pdflatex as default. You can delete it if not appropriate.  %%
%%=========================================================================================%%

%%\documentclass[sn-basic]{sn-jnl}% Basic Springer Nature Reference Style/Chemistry Reference Style

%%Note: the following reference styles support Namedate and Numbered referencing. By default the style follows the most common style. To switch between the options you can add or remove “Numbered” in the optional parenthesis. 
%%The option is available for: sn-basic.bst, sn-chicago.bst%  
 
%%\documentclass[pdflatex,sn-nature]{sn-jnl}% Style for submissions to Nature Portfolio journals
%%\documentclass[pdflatex,sn-basic]{sn-jnl}% Basic Springer Nature Reference Style/Chemistry Reference Style
\documentclass[pdflatex,sn-mathphys-num]{sn-jnl}% Math and Physical Sciences Numbered Reference Style
%%\documentclass[pdflatex,sn-mathphys-ay]{sn-jnl}% Math and Physical Sciences Author Year Reference Style
%%\documentclass[pdflatex,sn-aps]{sn-jnl}% American Physical Society (APS) Reference Style
%%\documentclass[pdflatex,sn-vancouver-num]{sn-jnl}% Vancouver Numbered Reference Style
%%\documentclass[pdflatex,sn-vancouver-ay]{sn-jnl}% Vancouver Author Year Reference Style
%%\documentclass[pdflatex,sn-apa]{sn-jnl}% APA Reference Style
%%\documentclass[pdflatex,sn-chicago]{sn-jnl}% Chicago-based Humanities Reference Style

%%%% Standard Packages
%%<additional latex packages if required can be included here>

\usepackage{graphicx}%
\usepackage{multirow}%
\usepackage{amsmath,amssymb,amsfonts}%
\usepackage{amsthm}%
\usepackage{mathrsfs}%
\usepackage[title]{appendix}%
\usepackage{xcolor}%
\usepackage{textcomp}%
\usepackage{manyfoot}%
\usepackage{booktabs}%
\usepackage{algorithm}%
\usepackage{algorithmicx}%
\usepackage{algpseudocode}%
\usepackage{listings}%
%%%%

%%%%%=============================================================================%%%%
%%%%  Remarks: This template is provided to aid authors with the preparation
%%%%  of original research articles intended for submission to journals published 
%%%%  by Springer Nature. The guidance has been prepared in partnership with 
%%%%  production teams to conform to Springer Nature technical requirements. 
%%%%  Editorial and presentation requirements differ among journal portfolios and 
%%%%  research disciplines. You may find sections in this template are irrelevant 
%%%%  to your work and are empowered to omit any such section if allowed by the 
%%%%  journal you intend to submit to. The submission guidelines and policies 
%%%%  of the journal take precedence. A detailed User Manual is available in the 
%%%%  template package for technical guidance.
%%%%%=============================================================================%%%%

%% as per the requirement new theorem styles can be included as shown below
\theoremstyle{thmstyleone}%
\newtheorem{theorem}{Theorem}%  meant for continuous numbers
%%\newtheorem{theorem}{Theorem}[section]% meant for sectionwise numbers
%% optional argument [theorem] produces theorem numbering sequence instead of independent numbers for Proposition
\newtheorem{prop}[theorem]{Proposition}% 
%%\newtheorem{proposition}{Proposition}% to get separate numbers for theorem and proposition etc.

\theoremstyle{thmstyletwo}%
\newtheorem{example}{Example}%
\newtheorem{remark}{Remark}%
\newtheorem{lem}{Lemma}

\theoremstyle{thmstylethree}%
\newtheorem{definition}{Definition}%

\raggedbottom
%%\unnumbered% uncomment this for unnumbered level heads
\newcommand{\pbar}{\ensuremath{\underline{p}}}

%\usepackage{arydshln}
\usepackage{float}
\usepackage{rotating}
%\usepackage{boldline}
\usepackage{array}

%\usepackage{multicol,wrapfig,multirow}

%\usepackage{natbib,amssymb,amsmath,amsthm,latexsym,makeidx,lscape,epic,overpic,graphicx,graphics,bm,pdflscape,rotating,multirow,bigdelim,color,epsfig,pgfpages} \usepackage{mathpazo}
%\usepackage[english]{babel}
%\usepackage[T1]{fontenc} \usepackage[left]{eurosym}
%\usepackage{pdfsync,epstopdf}
%\usepackage{multicol,wrapfig,multirow}
%\usepackage{epstopdf}
%\usepackage[draft]{hyperref}
%\usepackage{subcaption}
%\usepackage{indentfirst}
%\usepackage{tabularx, booktabs}
%\usepackage{float}
%\usepackage{rotating}
%\usepackage{ctable}
%\usepackage{arydshln}
%\usepackage{boldline}
%\setcitestyle{aysep={}} 

\makeatletter
\def\@cline#1-#2\@nil{...}
\makeatother


\begin{document}

\title[Reduction in Information Asymmetries]{Does the blockchain technology help to reduce information asymmetries?}

%%=============================================================%%
%% GivenName	-> \fnm{Joergen W.}
%% Particle	-> \spfx{van der} -> surname prefix
%% FamilyName	-> \sur{Ploeg}
%% Suffix	-> \sfx{IV}
%% \author*[1,2]{\fnm{Joergen W.} \spfx{van der} \sur{Ploeg} 
%%  \sfx{IV}}\email{iauthor@gmail.com}
%%=============================================================%%

%\author*[1,2]{\fnm{First} \sur{Author}}\email{iauthor@gmail.com}
%
%\author[2,3]{\fnm{Second} \sur{Author}}\email{iiauthor@gmail.com}
%\equalcont{These authors contributed equally to this work.}
%
%\author[1,2]{\fnm{Third} \sur{Author}}\email{iiiauthor@gmail.com}
%\equalcont{These authors contributed equally to this work.}
%
%\affil*[1]{\orgdiv{Department}, \orgname{Organization}, \orgaddress{\street{Street}, \city{City}, \postcode{100190}, \state{State}, \country{Country}}}
%
%\affil[2]{\orgdiv{Department}, \orgname{Organization}, \orgaddress{\street{Street}, \city{City}, \postcode{10587}, \state{State}, \country{Country}}}
%
%\affil[3]{\orgdiv{Department}, \orgname{Organization}, \orgaddress{\street{Street}, \city{City}, \postcode{610101}, \state{State}, \country{Country}}}

\author*[1]{\fnm{Papatya} \sur{Duman}}\email{Papatya.Duman@uni-bielefeld.de}

\author[2]{\fnm{Claus-Jochen} \sur{Haake}}\email{cjhaake@wiwi.upb.de}

\author[2]{\fnm{Simon} \sur{Hemmrich}}\email{Simon.Hemmrich@uni-paderborn.de}

\author[2]{\fnm{Alexander} \sur{Koch}}\email{Alexander.Koch@uni-paderborn.de}

\author[2]{\fnm{Sarah} \sur{K\"uhn}}\email{Sarah.Kuehn@uni-paderborn.de}

\author[2]{\fnm{Daniel} \sur{Beverungen}}\email{Daniel.Beverungen@uni-paderborn.de}


\affil*[1]{\orgdiv{Center for Mathematical Economics}, \orgname{Bielefeld University}, 
	\orgaddress{\street{Universitatsstr. 25}, \city{Bielefeld}, \postcode{33615}, \country{Germany}}}

\affil[2]{\orgname{Paderborn University}, 
	\orgaddress{\street{Warburger Str. 100},\city{Paderborn}, \postcode{33098}, \country{Germany}}}

%%==================================%%
%% Sample for unstructured abstract %%
%%==================================%%

\abstract{We examine the problem faced by a buyer seeking to purchase an experience good
	without prior knowledge of its stochastic quality. An expert who owns the
	product can be paid to provide a signal about its quality. Our analysis
	explores the impact of introducing a credible signaling mechanism for the
	buyer. Specifically, we propose using blockchain technology, which ensures
	immutability, decentralization, privacy, and transparency, to store the
	signal. Our findings reveal that this approach reduces the number of possible
	equilibria while preserving the ``good equilibrium'', in which information is both
	acquired and accurately transmitted. Consequently, the use of blockchain
	technology mitigates the equilibrium coordination problem and improves the
	provision of credible information.}

%%================================%%
%% Sample for structured abstract %%
%%================================%%

% \abstract{\textbf{Purpose:} The abstract serves both as a general introduction to the topic and as a brief, non-technical summary of the main results and their implications. The abstract must not include subheadings (unless expressly permitted in the journal's Instructions to Authors), equations or citations. As a guide the abstract should not exceed 200 words. Most journals do not set a hard limit however authors are advised to check the author instructions for the journal they are submitting to.
% 
% \textbf{Methods:} The abstract serves both as a general introduction to the topic and as a brief, non-technical summary of the main results and their implications. The abstract must not include subheadings (unless expressly permitted in the journal's Instructions to Authors), equations or citations. As a guide the abstract should not exceed 200 words. Most journals do not set a hard limit however authors are advised to check the author instructions for the journal they are submitting to.
% 
% \textbf{Results:} The abstract serves both as a general introduction to the topic and as a brief, non-technical summary of the main results and their implications. The abstract must not include subheadings (unless expressly permitted in the journal's Instructions to Authors), equations or citations. As a guide the abstract should not exceed 200 words. Most journals do not set a hard limit however authors are advised to check the author instructions for the journal they are submitting to.
% 
% \textbf{Conclusion:} The abstract serves both as a general introduction to the topic and as a brief, non-technical summary of the main results and their implications. The abstract must not include subheadings (unless expressly permitted in the journal's Instructions to Authors), equations or citations. As a guide the abstract should not exceed 200 words. Most journals do not set a hard limit however authors are advised to check the author instructions for the journal they are submitting to.}

\keywords{Blockchain, Signaling, Asymmetric Information, Coordination Problem}

\pacs[JEL Classification]{C72, D82, D47}

%%\pacs[MSC Classification]{35A01, 65L10, 65L12, 65L20, 65L70}

\maketitle

\section{Introduction}



Availability of correct information is one of the central aspects in economic
interaction. In his seminal paper, \cite{akerlof1978market}
demonstrates in a simple model for the market for lemons that asymmetric
information leads to market failure. In essence, two agents want to trade a
product, but the buyer does not have sufficiently accurate information on the
product quality, and hence on personal utilities. 
%When the traded good is an
%experience good, the true quality can only be observed as early as it is
%purchased and used.

The analysis of trade under asymmetric information is at the heart of contract
theory which aims at designing mechanisms that incentivize agents to reveal
their private information through taking specific observable actions. For
example, {\em signaling mechanisms}, as introduced by \cite{spence1973job},
allow the informed agent to send a message or signal to the uninformed
one. Although it cannot be verified directly that the agent submits the
credible signal, signaling costs are set up in a way that it is in the agent's
best interest not to misrepresent information. In practice, therefore,
costless signals only provide limited advice to potential buyers, e.g.,
(online) rating systems that are almost costless to use
\citep{dellarocas2003digitization}. Similarly, the problem of limited
informativeness occurs when the costs of a signal is either
independent from the signal itself or information on the signal costs cannot
credibly be shared. As a result, signaling games typically show a large number
of (separating and pooling) equilibria.\footnote{For our context,
	\cite{RePEc:mit:worpap:496} show that the notions of Perfect Bayesian
	Equilibrium and Sequential Equilibrium are equivalent and
	\cite[p.~103]{bolton2004contract}. state: ``In signaling games, the
	difficulty is not to find a PBE. Rather the problem is that there are too
	many PBEs.}

As an illustration of this problem we return to (online) rating systems
that are supposed to inform potential buyers about the quality of a
product. Typically, the costs of releasing a rating are low and not connected
to the product quality so that we observe more and more the emergence of fake
reviews \citep{he2022market} jeopardizing its original intent \citep{tadelis2016reputation}. From a technical perspective \cite{mohawesh2021fake} survey the
possibilities of detecting fake reviews by analyzing review texts. However,
such methods are designed for detection, not prevention. 


From a general perspective, a part of the problem is that information (e.g., on
product quality) and action (e.g., give a rating) are decoupled. A device like blockchain\footnote{This technology emerged
	from previous research on digital signatures \citep{rivest1978}, the Merkle
	tree \citep{merkle1980protocols}, cryptographically secure time-stamping with
	blocks \citep{haber1991time}, and the precursors of digital money
	\citep{dai1998bmoney, back2002hashcash}. Beyond its use as an infrastructure
	for cryptocurrencies, such as bitcoin, built-in features in the blockchain
	technology include immutability, decentralization, privacy and transparency,
	and distributed trust \citep{nakamoto, swan2015blockchain,
		defilippi2016interplay, seidel2018questioning}.
} provides an interesting possibility to link both, information and action, by storing
actions (in particular payments revealing information on incentives) in a way that this
information is traceable, public, immutable, and a trusted third party is not
required as intermediary \citep{franke2023limited}. Moreover, acquisition of
information stored in the blockchain can be designed costly, so that it is part
of a buyer's strategic choice to acquire a signal or not \citep{xu2023optimal,
	ZHANG2024126}.


In this paper, we investigate, whether the blockchain technology (BCT) can be used to reduce
information asymmetries in economic transactions. More precisely, we employ a
simple model in which a {\em buyer} wants to buy a product of unknown
quality. Assuming that the good is an experience good, its true quality cannot
be observed prior to purchase. However, the buyer may receive information from
another agent, named {\em expert}\footnote{Here, the term ``expert'' refers
	to an agent, who has already bought the product and has therefore experience
	with it and can assess its quality.}, who has already bought the product and
therefore is informed about the true product quality. In a ``traditional modeling'',
the buyer may pay the expert for a quality assessment, and then base
his\footnote{Throughout this text, we use 'he' and 'she' to differentiate
	between two types of agents (expert - she, buyer - he). This choice of
	pronouns does not imply any gender-specific characteristics or biases
	regarding the individuals discussed.}  buying decision on the expert's
signal. In this model, the expert's costs of providing any signal are
independent of the signal. As we will demonstrate later (Proposition
\ref{prop:1}), the corresponding two-person game exhibits multiple sequential
equilibria in pure strategies, so that a severe coordination problem occurs.

We compare these equilibria with those in a model, in which BCT is available. More precisely, the seller, who is not modeled as an
active agent, allows the expert to buy the product at either a regular or a
discounted price.\footnote{We may also think of the expert as the first buyer
	of the product.} The expert may decide which price to pay for the purchase
after experiencing the product quality. Her payment information is stored in a
blockchain. It follows that the signal she sends has a direct impact on her
overall payoff. Phrased differently, the cost of a signal depends on the
signal. Instead of ``just asking'' the expert about the product quality, the
buyer now has the possibility to pay for observing the stored transaction
between the expert and the seller. In particular, signaling high quality (via
paying the regular price) now comes at a higher cost for the expert, thus
increases her signal's credibility \citep{pornpitakpan2004persuasiveness}.

The direct implication is that the existence of a device that stores
immutable information, which is observable at a given cost, helps to reduce
a coordination problem between an expert and an uninformed potential buyer. The
``good equilibrium'', i.e., the expert always pays the price that matches the
quality of the product, and the buyer acquires information, then perfectly
correlates his buying decision on the signal, is still an equilibrium, however
reachable with fewer coordination effort. In
the game induced in the model with BCT, we show that there
are fewer sequential equilibria compared to the game without BCT (Proposition
\ref{prop:2}).

As one application, a trading system with experts that are offered special
purchase conditions can be designed to better inform buyers about product
qualities. From an information economics perspective, such a system works
because the cost of a signal becomes observable, and as a result signal
credibility is enhanced.  Many online platforms feature reputation systems to
reduce uncertainty in transactions \citep{rice} and promote trust in sellers
\citep{morenoTerwiesch}. The more reliable the implemented reputation
mechanism is, the more it enhances trust in buying decisions
\citep{honhonHyndman,josangetal}.
Still, reputation systems are prone to strategic manipulation, especially due
to false quality signals induced by low-quality sellers \citep{bolton2004how,
	bolton2013engineering, kennesSchiff}.

Implementing reputation systems on a blockchain is considered the most reliable way to store feedback information and exclude manipulation \citep{bellinietal, caiZhu,hawlitscheketal}.
A blockchain is a tamper-proof, chronological ledger implemented as a distributed database for transactions secured by cryptographic mechanisms and governed by a consensus protocol \citep{becketal}. A community of computing nodes produces a blockchain by exchanging and storing digital transactions that are recorded in blocks, building an ever-growing chain. Cryptographic mechanisms guarantee that transactions can be verified and any unauthorized change of transaction data is easy to detect. The consensus protocol provides economic incentives that ensure only valid transactions are stored in the blockchain. In a truly decentralized network, distributed consensus mechanisms provide an extraordinary degree of tamper-resistance, removing the necessity to include a trusted third party for checking transactions' legitimacy and authenticity \citep{nakamoto,tschorschscheuermann}.

From the contract theory perspective, \cite{bolton2004contract},
\cite{kreps1994signalling}, or \cite{10.1257/jel.52.4.1075} provide extensive
survey on how to incentivize agents to reveal private information. As one
expects, there remains a trade-off between incentive compatibility and
efficiency.

\cite{MAMADA2022107} follows an interesting approach to solve the market for
lemons problem by considering signal indices already defined in \cite{spence1973job} that can costly be observed. For
used cars this means that the state of air-conditioning can costly be
verified, which is correlated with the quality of the car. He shows that if
the price falls into a specific range, cars with bad air-conditioning are
driven out of the market. Unlike in our approach, there has to be an
opportunity to test the product prior to purchase.

\cite{GERSBACH202230} discuss a model
with several low and high quality sellers and one buyer, who potentially wants
to buy more than one product. They set up a four stage mechanism, in which
prices depend on the actions of all sellers and show that in equilibrium all
high-quality sellers send a different signal than low-quality
sellers. Compared to our work, we focus on the revelation of quality
information through the interaction of (two) buyers.

In supply chain manufacturing, a blockchain can be used to store the quality
of a product itself. In such models, information disclosure through BCT is a
strategic choice and will only be used by high-quality sellers
\citep{ZHANG2024126,xu2023optimal}. However, in
our model, product quality is not verifiable prior to purchase and cannot
credibly be shared, so that the agents are dependent on quality signals. 


The paper is organized as follows: Section~\ref{sec:model1} introduces the two
games with imperfect information without (Model 1) and with (Model 2) deploying BCT. While Section~\ref{sec:analysis} briefly reviews the concept of
sequential equilibria, the results are presented in
Section~\ref{sec:results}. Section~\ref{sec:discussion} concludes. All
calculations of equilibria are relegated to the appendices.

\section{Model}
\label{sec:model}

In this section, we set up two models describing the interaction between two
buyers of the same product, the quality of which is a priori unknown. 
More precisely, a seller sells a product at a given unit price $p>0$
and delivers the product in either good or bad quality. As we are primarily
interested in mechanisms for information transmission between subsequent
buyers, the product quality is not a strategic choice of the seller, but the
result of a chance move with the known priors $q$ for {\em high} quality and
$1-q$ for {\em low} quality. Alternatively, on e may think of $q$ ($1-q$) as
probabilities to trade with a high (low) quality seller.

Further, we assume that the product is an experience good meaning that a buyer
can observe its quality after consumption, i.e., not before a purchase. Each
buyer has use for one unit only, so that the buying decision is not about the
quantity. In case of a purchase, the buyer's utility from consumption
expressed as willingness to pay is $u>0$, when the product quality is high,
and $0$ otherwise.  In order to reduce information asymmetries, the buyer can
retrieve information on the product quality from an expert (or experienced
buyer), who has already bought and assessed one unit of the same product. The expert's
utility from consumption is either $v>0$ or $0$ given that the quality is high
or low, respectively. Both, expert and buyer, are assumed to be risk-neutral.
To eliminate the effects of long-term reputation, we
think of a one-shot interaction taking place anonymously, e.g., on an online
platform.

In what follows, we introduce and compare two versions of information
transmission from the expert to the buyer. In both models, the sequence of
actions is described as a sequential move game with imperfect
information. First the expert buys the product and is therefore perfectly
informed about the product quality, i.e., realization of the chance
move. After that the buyer enters the scene, who has no quality information,
but knows that the expert has observed the quality. 

We analyze two mechanisms to cope with this information asymmetry. Before
buying the product the buyer may acquire a signal on product quality from the
expert. The difference between the models lies in how information is signaled
and how these signals are designed. In both models, there will be an incentive
for the expert to report truthfully. In the first model the expert's signal is
her (non-verifiable) report on the quality and the buyer has to completely
rely on it. in the second model, the expert is offered the possibility to pay
$\pbar < p$ that can serve as a signal for low quality. The actual
payment, however, is not connected to the true quality and solely the expert's
choice. In the second model we use BCT to store the expert's signal, i.e., the
real payment. This way the signal is confirmed, immutable and can be retrieved by
the (any) buyer.

Our analyses focuses on the sequential equilibria of the games in both
models. More precisely, we investigate how the set (and number) of
equilibria changes when we move from the first to the second model, i.e., when
BCT is deployed.

\medskip

\subsection*{\bf  Model 1: Reported quality}
%{\noindent\bf Model 1: Reported quality}
\label{sec:model1}

Figure~\ref{fig:Model1.1} illustrates the game tree in Model~1. The game
starts with a chance move with probabilities $q$ and $1-q$ choosing high or
low quality, respectively. We assume that the expert then buys the product at
price $p$ and directly assesses its quality, i.e., she observes the outcome of
the chance move. Depending on whether the true quality is high or low, the
expert makes a decision on which quality level to report to the buyer when
asked.\footnote{To have a greater similarity of the game trees of the two
	models, this decision is modeled prior to the buyer's decision whether
	or not to acquire information from the expert.} More
precisely, given the true quality is ``high'' (``low''), the expert either
chooses $H^+$ or $L^+$ ($H^-$ or $L^-$) to signal high or low
quality.\footnote{While $H$ and $L$ refer to the experts action, the
	superscript ``$+$'' (``$-$'') means that the true quality is ``high''
	(``low'').}

From now on, all remaining decisions are taken by the uninformed buyer.  He
first has the option to acquire ({\em ac}) information from the expert by
paying her an amount $x>0$ or refuse ({\em ref}) to do so.  After that, he
takes a final decision on buying or not buying the product.  Formally, he
chooses an action {\em buy$^+$}, {\em buy$^-$}, or {\em buy$^0$} for buying
the product, after receiving a signal of \textit{high quality}, \textit{low
	quality}, or \textit{no signal on quality}, respectively. Similarly,
the actions {\em nb$^+$}, {\em nb$^-$}, or {\em nb$^0$} refer to not buying
the product.\footnote{Here superscripts do not refer to the true quality of 
	the product but to the information provided (``$+$'' for ``high'' signal, 
	``$-$'' for ``low'' signal) or not provided (``$0$'' for no signal) by the expert.}
Due to the unobservability of the chance move and possibly unacquired
information, the buyer has four information sets, displayed in different
colors.

\begin{figure}[]
	\centering
	\includegraphics[scale=0.55]{NoBCModel}
	\caption{Game tree in Model~1: Reported quality	}
	\label{fig:Model1.1}
\end{figure}



At each terminal node, the payoffs for expert (top line) and buyer (bottom
line) are displayed. They are determined as follows: Independent of the
product quality, the expert pays the price $p$. Her net utility is $v-p$ when
the realization of the chance move is {\em high}, and $0-p$ otherwise. If the
buyer buys the product, he pays the same price $p$. In case the bought
product is of high quality, he receives $u$, and $0$ for a low quality
product. If he does not buy the product, his utility is $0$.  Whenever the
buyer chooses {\em ac} and therefore the expert sends a quality signal, a
fee $x >0$ is transferred from the buyer to the expert.

So far, the expert is not provided with incentives to tell the truth, because
the fee $x$ she may receive is independent of the signal she gives. In addition,
the expert receives a bonus $r>0$ paid by the platform, when the buyer
bought the product and the report was truthful. We may think of the buyer
approving or not the acquired information, but do not model it as an explicit
action.\footnote{As the game is over after the approval, it is not necessary
	to incentivize the buyer to tell the truth here.}  The platform
(owner) is not explicitly modeled as an agent. As the platform is interested in having a working system
that allows buyers to acquire information from the expert and receive a
correct response, there is good reason to pay the bonus $r$. 
From a far-sighted perspective, such a possibility can
attract future buyers to use the platform.


In sum, the payoffs depend on the utility received from a
high quality product ($u$ or $v$), the price of the product $p$, and possibly
on the fee $x$ and bonus $r$. The following assumptions capture specific
scenarios for which we will analyze equilibria in the next section.



\begin{description}
	\item[${\bf A1}$ ] $uq - p > 0 \qquad $ and $\qquad vq - p > 0$
	\item[\textbf{${\bf A1' }$}] $uq - p < 0 \qquad $ and $\qquad vq - p > 0$ 
	\item[$ {\bf A2 }$] $(1-q)p > x$
	\item[${\bf A2' }$] $q(u-p) > x $
\end{description}

Assumptions $ A1$  and $  A1'$ are mutually exclusive. $ A1 $ guarantees that the expected
utility from buying the product exceeds the price for the buyer. It can be viewed as a participation constraint and implies that a
completely uninformed, risk-neutral buyer does buy the
product.\footnote{Technically, it implies that the action $buy^0$ dominates
	the action $nb^0$ in the corresponding (blue) information set.} In contrast,
assumption $ A1' $ requires the buyer not buy the product, when he is
uninformed. In either assumption $ A1 $ and $ A1' $, the buyer expects to have a
positive net utility from buying the product ($ u-p>0 $).

Assumptions $ A2 $ and $ A2' $ relate the fee $x$ to the buyer's expected net utility or
net loss. Assumption $ A2 $ can be rewritten to
$q(u-p)+(1-q)(0-p) < q(u-p)- x$, which means that the expected utility from buying
the product without information is lower compared to having the correct signal
at cost $x$. Phrased differently, it ensures that paying the fee is better
than receiving a low quality product. Otherwise, there would be no incentives
to ask an expert. Assumption $ A2' $ ensures that the fee is smaller than the
expected utility gain, so that paying $x$ is overcompensated by
receiving a high quality product.
Interestingly, when subtracting the l.h.s.~in $ A2 $ from the l.h.s.~in $ A2' $ we get
$q(u-p) - (1-q)p = qu-p$ which is positive (negative) under $ A1 $ ($ A1' $). A direct
implication is given in the following lemma.$  $

\begin{lem}
	Assumption $A1$ and $A2$ together imply $A2'$. Assumptions $A1'$ and
	$A2'$ jointly imply $A2$. 
\end{lem}
% \begin{lem}
	% 	Under assumption $ A1 $, $ A2 $, the assumption $ A2' $ is implied.   Under assumption $ A1' $, $ A2' $ the assumption $ A2 $ is implied. 
	% \end{lem}



\begin{figure}[]
	\centering
	\includegraphics[scale=0.55]{BCModel}
	\caption{Game tree in Model~2: Information access via blockchain
	}
	\label{fig:Model2.1}
\end{figure}


In essence the key characteristic of Model~1 lies in the fact that the buyer can choose to base his
buying decision on information that cannot credibly be transmitted from the
expert to the buyer, as he cannot learn the result of the chance move without
buying the product.



\medskip 


% \newpage
\subsection*{\bf  \bf Model 2: Information access via blockchain}
%{\noindent\bf Model 2: Information access via blockchain}
\label{sec:model2}



The game tree in Model~2 is depicted in Figure~\ref{fig:Model2.1}. What
changes is the type of signal from the expert and the access to information.
We now assume that a two-price contract is offered from the seller to the
expert. That means, she buys the product, assesses its quality, and can then
either pay the regular price $p$ (action {\em reg}) or a specified discounted
amount $\pbar < p$ (action {\em dis}). The final payment to the seller is
stored in a blockchain so that it cannot be modified anymore and can be
retrieved from a third party. This stored information is used as the signaling
feature of the model. The difference to Model~1 is that the price paid by the
expert serves as the signal. Hence, signaling high quality by paying the
regular price instead of a discounted one comes at a cost for the expert. This
is in line with the general idea of signaling information in economics. The
more costly a signal is, the more credible it is
\citep{pornpitakpan2004persuasiveness}. In this model, signaling the high
quality by paying the higher price preserves the chance of receiving the bonus
payment $ r $ at the end.




The buyer chooses to acquire information (action {\em ac}) or not (action {\em
	ref}) from the blockchain at a fee $x$ paid to the expert. In the former
case, he observes whether the expert made the regular or the discounted
payment.  A regular payment (discounted payment) of $p$ ($\pbar$) is supposed
to signal high (low) quality. But information on payment can credibly be
obtained thanks to the use of a blockchain.

The sequence of the buyer's actions is identical to the first model, so we
keep the same notation for actions and payoffs. The main difference in the
payoffs is that the expert can choose to pay either $p$ or $\pbar$.  Again,
at the very end the expert may receive a bonus $r$, if the buyer 
acquired information, bought the product, and approved accuracy of the signal.

Additional to Assumptions $A1$-$A2$ (or $A1'$-$A2'$), it is instructive to
introduce the following intuitive assumption on the bonus payment $r$:

\begin{description}
	\item[ ${\bf A3 }$] $r > p - \pbar$.
\end{description}

Assumption $ A3 $ guarantees that it is more attractive to pay the regular price
and receive the bonus $ r $ than to pay the discounted price. Otherwise, the
expert has no incentive to pay the regular price $p$.
\section{Strategic Behavior}
\label{sec:analysis}


To economize on notation, the expert and the buyer are also denoted as agent 1
and 2, respectively. A {\em pure strategy} for agent $i$ (i=1,2) is a mapping
that takes each of agent $i$'s information sets to one of the feasible actions
therein.  In Model 1, we can therefore describe the expert's possible pure
strategies by
$S^{1,1} := \left\{ H^+, L^+ \right\} \times \left\{ H^-, L^- \right\}$. In
Model 2, her pure strategies are given by
$S^{2,1} := \left\{ reg^+, dis^+ \right\} \times \left\{ reg^-, dis^-
\right\}$. The set of the buyer's pure strategies is the same in both models
and given by
$S^2 = S^{1,2} = S^{2,2} := \left\{ ac, ref \right\} \times \left\{ buy^+,
nb^+ \right\} \times \left\{ buy^-, nb^- \right\} \times \left\{ buy^0, nb^0
\right\}$.

We analyze agents' strategic behaviors by inspecting the extensive form games
with imperfect information as described in Model~1 and 2. From a static
viewpoint, the agents choose their plans of action prior to the game and the
game play results from execution of these plans.  In the normal-form game the
strategy sets are the pure strategies as above and payoffs at a strategy
profile are formed by following the path that is visited by the application of
the actions in the strategy profile. This way we can identify Nash equilibria (NE)
of the normal-form game. However, this static view neither takes the
sequentiality of decisions nor imperfect information into account. Moreover,
the number of Nash equilibria is typically large, so a refinement of the Nash
equilibrium concept strengthens the predictive power and typically reduces the
number of equilibria. Yet, as both versions of the game (from the two models)
admit only a single subgame, the subgame perfect equilibrium concept has no
further bite compared to Nash equilibria (of the normal-form game).
We therefore discuss the
concept of sequential equilibria introduced by \cite{krepswilson82} next.

A {\em behavioral strategy $b_i$ for agent $i$} specifies a probability distribution over the feasible
actions in each information
set, in which agent $i$ chooses. A pure strategy of agent $i$ is a special behavioral strategy,
which assigns in each information set a probability of 1 to the action chosen
in the pure strategy (and hence 0 to all other actions). A behavioral strategy
of agent $i$ is {\em completely mixed} if it assigns a positive probability to
each of the feasible actions in each of an agent's information set.


A {\em belief} $\beta_i$ of agent $i$ includes for each information set of
that agent a probability distribution over the nodes included in the
information set. A pair $ \beta=(\beta_1,\beta_2) $ of beliefs for both agents is a {\em belief system}. A
tuple $(b,\beta)=((b_1,b_2),(\beta_1,\beta_2))$ consisting of a profile of
behavioral strategies and a belief system is termed an {\em assessment}.


To define sequential equilibria we first discuss the following properties for
an assessment $(b,\beta)$.
\begin{description}
	\item[\textbf{Sequential Rationality}] Within each information set $h_i$ of agent $i$,
	the probability distribution $b_i(h_i)$ over actions in $h_i$ maximizes the
	agent's expected payoff given the belief $\beta_i(h_i)$ over nodes in that
	information set and the behavioral strategy profile $b$.
	\item [\textbf{Bayesian Consistency}] The belief system $ \beta=(\beta_1,\beta_2) $ is
	generated by the strategy profile via Bayesian updating. That means, given
	the probabilities in $b=(b_1,b_2)$ to take actions, the belief of deciding
	at a specific node in information set $h$ is the probability of
	reaching that node conditional on reaching some note in $h$.
\end{description}


Sequential rationality requires the agents choose behavioral strategies that
maximize their payoffs given their beliefs about the node at which the
decision has to be taken. Bayesian updating means that given the
strategy choices, agents' beliefs are formed in a compatible way. However, it
could happen that by the application of a behavioral strategy profile, an
information set is not visited at all. In those information sets $h$, the
belief formation via Bayesian updating is not possible. It means that a
rationality condition based on Bayesian updating per se makes no sense, so
that $b_i(h)$ cannot be identified as an optimal choice in $h$. For such cases, we need a more sophisticated form of consistency.

\begin{description}
	\item [\textbf{Consistency}] There exists a sequence
	$ (b^m,\beta^m)_{m \in \mathbb{N}} $ of assessments in which $ b^m $ is
	completely mixed and $\beta^m$ is Bayesian consistent so that
	$\lim_{m \longrightarrow \infty}(b^m,\beta^m)=(b,\beta) $ holds.
\end{description}


{\noindent Now, we are ready to define our equilibrium concept:}
\begin{description}
	\item [\textbf{Sequential Equilibrium}] A \textit{sequential
		equilibrium (SE)} is an
	assessment $ (b, \beta) $ which satisfies Sequential Rationality and
	Consistency.
\end{description}



If the behavioral strategies are
such that all information sets in the game tree are reached with positive
probability, 
Bayesian consistency and consistency coincide. Hence, consistency is trivially satisfied. Then in equilibrium, sequential rationality and Bayesian
consistency ensure optimal decisions w.r.t.~correct beliefs. If an information set $h$ is reached
with zero probability, belief formation according to Bayesian updating as well
as rational choice is not possible. Nonetheless, the equilibrium action choice in $h$ is
rationalizable, if it maximizes the agent's payoff given plausible beliefs,
i.e., beliefs $ (\beta^m)_{m \in \mathbb{N}} $ which are formed via behavioral strategies $ (b^m)_{m \in \mathbb{N}} $ being close to the
equilibrium strategies $ b $. Completely mixed strategies $ b^m$ for any $ m\in \mathbb{N} $ guarantee
that every information set is reached with positive probability, so that the
plausible beliefs $ \beta^m $ stem from Bayesian updating.



When comparing sequential equilibria in the extensive form game with (pure)
Nash equilibria of the normal-form game, it is easy to see that any sequential
equilibrium is a Nash equilibrium, which is due to sequential
rationality.\footnote{Sequential rationality means that deviating to another
	strategy reduces the payoff, as it is not optimal in one information
	set.} Conversely, not every Nash equilibrium needs to be a sequential
one. Intuitively, a Nash equilibrium may include actions that are not
sequentially rational in an information set as a ``threat'' to make the
opponent stay away from a particular strategy choice and not reach that
information set. Therefore, the sequential equilibrium concept is insensible to this form
of non-credible threats. Roughly speaking, increasing the degree of
rationality built in the equilibrium concept reduces the number of equilibria.

\section{Results}
\label{sec:results}

The ``desired'' or ``good'' behavioral strategy is without a doubt the one in which
the expert reports or pays according to the true quality, then the buyer
acquires information and buys according to the signal he receives. The buying
decision when no information is available is governed by Assumption $A1$ (buy)
or $A1'$ (not buy). Let us denote this behavioral strategy by
$b^{des}$. Besides the question, whether $b^{des}$ is part of some sequential
equilibrium (together with a consistent belief system) we are interested in
finding out whether the use of BCT reduces information
asymmetries. We use the following route to answer this second question: For
the games in Model 1 and in Model 2, we compare the numbers (and sets) of
sequential equilibria, i.e., investigate if employing the blockchain method
helps to reduce coordination issues. Attaining uniqueness of the equilibrium
in Model 2 is certainly to much to hope for, because the strategy profile, in
which the expert never pays the regular price and the buyer does not acquire
information, will always form a sequential equilibrium. The expert cannot
improve her payoff, because she can never receive the bonus $r$ and the buyer
cannot be better off as the expert's strategy choice does not reveal any
information.

To compare numbers of equilibria, we restrict attention to pure sequential
equilibria, as there are at most finitely many.\footnote{That means, each
	equilibrium satisfies the requirements of a sequential equilibrium from the
	previous section. In particular, there is no mixed sequential equilibrium
	that provides a greater payoff to some agent given the equilibrium beliefs.}
The sequential equilibrium concept is a refinement of the Nash equilibrium
concept applied to the normal-form derived from the sequential move
game(s). To start, there is a large number of pure Nash equilibria. The
game in Model 1 has 10 pure Nash equilibria with 2 different equilibrium
outcomes\footnote{The outcome of an assessment is a probability distribution
	over the terminal nodes of the game tree.}. In Model 2 (with BCT) we have 6 Nash equilibria in pure strategies,  again with
2 different outcomes. For details see Table~\ref{Seq.eq.:both} and for the computation see \ref{sec:seq}.

%\renewcommand{\arraystretch}{1.3}
%\begin{table}[h]
%	\centering
%	\begin{tabular}{ |p{1cm}||p{1.2cm}:p{1.2cm}|p{2.3cm}|p{1.2cm}|p{1.2cm}|p{1.2cm}|p{1.2cm}|p{1.2cm}| }
%		\hline
%		\multicolumn{8}{|c|}{Under the assumptions $ A1, A2$, and $ A3 $} \\
%		\hline
%		&\multicolumn{2}{c|}{Player 1} & \multicolumn{1}{c|}{Player 2}  & \multicolumn{2}{c|}{NE} &\multicolumn{2}{c|}{SE}\\
%		\cmidrule{2-8} & Model 1 & Model 2  & \multicolumn{1}{c|}{Model 1 and 2} & \multicolumn{1}{c|}{Model 1} & \multicolumn{1}{c|}{Model 2} & \multicolumn{1}{c|}{Model 1} & \multicolumn{1}{c|}{Model 2}\\
%		\hline
%		\textcolor{blue}{I ($b^{des}$)}   & \textcolor{blue}{$H^+L^-$} & \textcolor{blue}{$reg^+dis^-$} & \textcolor{blue}{$ac~buy^+nb^-buy^0$}& \multicolumn{1}{c|}{\textcolor{blue}{$ \checkmark $}} & \multicolumn{1}{c|}{\textcolor{blue}{$ \checkmark $}} & \multicolumn{1}{c|}{\textcolor{blue}{$ \checkmark $}} & \multicolumn{1}{c|}{\textcolor{blue}{$ \checkmark $}}\\
%		II &  $H^+L^-$ & $reg^+dis^-$ & $ac~buy^+nb^-nb^0$   & \multicolumn{1}{c|}{$ \checkmark $}& \multicolumn{1}{c|}{$ \checkmark $} & \multicolumn{1}{c|}{$ \times $} & \multicolumn{1}{c|}{$ \times $}\\
%		\textcolor{blue}{III} &\textcolor{blue}{$L^+ L^-$} & \textcolor{blue}{$dis^+dis^-$} & \textcolor{blue}{$ref~buy^+buy^-buy^0$}& \multicolumn{1}{c|}{\textcolor{blue}{$ \checkmark $}} & \multicolumn{1}{c|}{\textcolor{blue}{$ \checkmark $}}  &  \multicolumn{1}{c|}{\textcolor{blue}{$ \checkmark $}} & \multicolumn{1}{c|}{\textcolor{blue}{$ \checkmark $}}\\
%		\textcolor{blue}{IV}    &\textcolor{blue}{$L^+L^-$} & \textcolor{blue}{$dis^+dis^-$} & \textcolor{blue}{$ref~nb^+buy^-buy^0$}& \multicolumn{1}{c|}{\textcolor{blue}{$ \checkmark $}} & \multicolumn{1}{c|}{\textcolor{blue}{$ \checkmark $}}  &  \multicolumn{1}{c|}{\textcolor{blue}{$ \checkmark $}} & \multicolumn{1}{c|}{\textcolor{blue}{$ \checkmark $}}\\
%		V&   $L^+ L^-$ & $dis^+dis^-$ & $ref~nb^+nb^-buy^0$& \multicolumn{1}{c|}{$ \checkmark $}& \multicolumn{1}{c|}{$ \checkmark $} & \multicolumn{1}{c|}{$ \times $} & \multicolumn{1}{c|}{$ \times $}\\
%		VI& $L^+L^-$ & $dis^+dis^-$ & $ref~buy^+nb^-buy^0$ & \multicolumn{1}{c|}{$ \checkmark $}& \multicolumn{1}{c|}{$ \checkmark $} &\multicolumn{1}{c|}{$ \times $} & \multicolumn{1}{c|}{$ \times $}\\
%		VII& $H^+H^-$ & $reg^+reg^-$ & $ref~buy^+buy^-buy^0$& \multicolumn{1}{c|}{$ \checkmark $}& \multicolumn{1}{c|}{$ \times $} & \multicolumn{1}{c|}{$ \checkmark $} & \\
%		VIII& $H^+ H^-$ & $reg^+reg^-$  & $ref~nb^+nb^-buy^0$& \multicolumn{1}{c|}{$ \checkmark $}& \multicolumn{1}{c|}{$ \times $} & \multicolumn{1}{c|}{$ \times $} & \\
%		IX& $H^+H^-$  & $reg^+reg^-$ & $ref~buy^+nb^-buy^0$& \multicolumn{1}{c|}{$ \checkmark $}& \multicolumn{1}{c|}{$ \times $} & \multicolumn{1}{c|}{$ \checkmark $} &\\
%		X& $H^+ H^-$  & $reg^+reg^-$ & $ref~nb^+buy^-buy^0$& \multicolumn{1}{c|}{$ \checkmark $}& \multicolumn{1}{c|}{$ \times $} & \multicolumn{1}{c|}{$ \times $} & \\
%		\hline  \hline 
%		\multicolumn{8}{|c|}{Under the assumptions $ A1', A2'$, and $ A3 $} \\
%		\hline
%		XI  & $H^+L^-$ & $reg^+dis^-$ &$ac~buy^+nb^-buy^0$& \multicolumn{1}{c|}{$ \checkmark $} & \multicolumn{1}{c|}{$ \checkmark $} & \multicolumn{1}{c|}{$ \times $} & \multicolumn{1}{c|}{$ \times $}\\
%		\textcolor{blue}{XII} ($b^{des}$) &  \textcolor{blue}{$H^+L^-$} & \textcolor{blue}{$reg^+dis^-$} & \textcolor{blue}{$ac~buy^+nb^-nb^0$}   & \multicolumn{1}{c|}{\textcolor{blue}{$ \checkmark $}}& \multicolumn{1}{c|}{\textcolor{blue}{$ \checkmark $}} & \multicolumn{1}{c|}{\textcolor{blue}{$ \checkmark $}} & \multicolumn{1}{c|}{\textcolor{blue}{$ \checkmark $}}\\
%		XIII &$L^+ L^-$&$dis^+dis^-$ &$ref~buy^+buy^-nb^0$& \multicolumn{1}{c|}{$ \checkmark $} & \multicolumn{1}{c|}{$ \checkmark $} &  \multicolumn{1}{c|}{$ \times $} & \multicolumn{1}{c|}{$ \times $}\\
%		XIV    &$L^+L^-$ &$dis^+dis^-$ &$ref~nb^+buy^-nb^0$& \multicolumn{1}{c|}{$ \checkmark $} & \multicolumn{1}{c|}{$ \checkmark $}  &  \multicolumn{1}{c|}{$ \times $} & \multicolumn{1}{c|}{$ \times $}\\
%		\textcolor{blue}{XV}&   \textcolor{blue}{$L^+ L^-$} & \textcolor{blue}{$dis^+dis^-$} & \textcolor{blue}{$ref~nb^+nb^-nb^0$}& \multicolumn{1}{c|}{\textcolor{blue}{$ \checkmark $}}& \multicolumn{1}{c|}{\textcolor{blue}{$ \checkmark $}} & \multicolumn{1}{c|}{\textcolor{blue}{$ \checkmark $}} & \multicolumn{1}{c|}{\textcolor{blue}{$ \checkmark $}}\\
%		\textcolor{blue}{XVI}& \textcolor{blue}{$L^+L^-$} & \textcolor{blue}{$dis^+dis^-$} & \textcolor{blue}{$ref~buy^+nb^-nb^0$} & \multicolumn{1}{c|}{\textcolor{blue}{$ \checkmark $}}& \multicolumn{1}{c|}{\textcolor{blue}{$ \checkmark $}} &\multicolumn{1}{c|}{\textcolor{blue}{$ \checkmark $}} & \multicolumn{1}{c|}{\textcolor{blue}{$ \checkmark $}}\\
%		XVII& $H^+H^-$ & $reg^+reg^-$ & $ref~buy^+buy^-nb^0$& \multicolumn{1}{c|}{$ \checkmark $}& \multicolumn{1}{c|}{$ \times $} & \multicolumn{1}{c|}{$ \times $} & \\
%		XVIII& $H^+ H^-$ & $reg^+reg^-$  & $ref~nb^+nb^-nb^0$& \multicolumn{1}{c|}{$ \checkmark $}& \multicolumn{1}{c|}{$ \times $} & \multicolumn{1}{c|}{$ \checkmark $} & \\
%		XIX& $H^+H^-$  & $reg^+reg^-$ & $ref~buy^+nb^-nb^0$& \multicolumn{1}{c|}{$ \checkmark $}& \multicolumn{1}{c|}{$ \times $} & \multicolumn{1}{c|}{$ \times $} &\\
%		XX& $H^+ H^-$  & $reg^+reg^-$ & $ref~nb^+buy^-nb^0$& \multicolumn{1}{c|}{$ \checkmark $}& \multicolumn{1}{c|}{$ \times $} & \multicolumn{1}{c|}{$ \checkmark $} & \\
%		\hline
%	\end{tabular} 
%	\caption{Sequential equilibria of both models under the assumptions ${\bf A1, A2, A3 }$ or ${\bf A1', A2', A3 }$}
%	\label{Seq.eq.:both}
%\end{table}
\begin{table}[h]
	\centering
	\renewcommand{\arraystretch}{1.3}
	\begin{tabular}{p{1cm} p{1.2cm} p{1.2cm} p{2.3cm} p{1.1cm} p{1.1cm} p{1.1cm} p{1.1cm}}
		\toprule
		\multicolumn{8}{c}{Under the assumptions $ A1, A2$, and $ A3 $} \\
		\midrule
		& \multicolumn{2}{c}{Player 1} &\multicolumn{1}{c}{Player 2} & \multicolumn{2}{c}{NE} & \multicolumn{2}{c}{SE} \\
		\cmidrule(lr){2-3} \cmidrule{4-4} \cmidrule(lr){5-6} \cmidrule(lr){7-8}
		& Model 1 & Model 2 & Model 1 and 2 & Model 1 & Model 2 & Model 1 & Model 2 \\
		\midrule
		
		\textcolor{blue}{I ($b^{des}$)} & \textcolor{blue}{$H^+L^-$} & \textcolor{blue}{$reg^+dis^-$} & \textcolor{blue}{$ac~buy^+nb^-buy^0$}
		& \textcolor{blue}{$\checkmark$} & \textcolor{blue}{$\checkmark$} & \textcolor{blue}{$\checkmark$} & \textcolor{blue}{$\checkmark$} \\
		
		II & $H^+L^-$ & $reg^+dis^-$ & $ac~buy^+nb^-nb^0$
		& $\checkmark$ & $\checkmark$ & $\times$ & $\times$ \\
		
		\textcolor{blue}{III} & \textcolor{blue}{$L^+ L^-$} & \textcolor{blue}{$dis^+dis^-$} & \textcolor{blue}{$ref~buy^+buy^-buy^0$}
		& \textcolor{blue}{$\checkmark$} & \textcolor{blue}{$\checkmark$} & \textcolor{blue}{$\checkmark$} & \textcolor{blue}{$\checkmark$} \\
		
		\textcolor{blue}{IV} & \textcolor{blue}{$L^+L^-$} & \textcolor{blue}{$dis^+dis^-$} & \textcolor{blue}{$ref~nb^+buy^-buy^0$}
		& \textcolor{blue}{$\checkmark$} & \textcolor{blue}{$\checkmark$} & \textcolor{blue}{$\checkmark$} & \textcolor{blue}{$\checkmark$} \\
		
		V & $L^+ L^-$ & $dis^+dis^-$ & $ref~nb^+nb^-buy^0$
		& $\checkmark$ & $\checkmark$ & $\times$ & $\times$ \\
		
		VI & $L^+L^-$ & $dis^+dis^-$ & $ref~buy^+nb^-buy^0$
		& $\checkmark$ & $\checkmark$ & $\times$ & $\times$ \\
		
		VII & $H^+H^-$ & $reg^+reg^-$ & $ref~buy^+buy^-buy^0$
		& $\checkmark$ & $\times$ & $\checkmark$ & {} \\
		
		VIII & $H^+ H^-$ & $reg^+reg^-$ & $ref~nb^+nb^-buy^0$
		& $\checkmark$ & $\times$ & $\times$ & {} \\
		
		IX & $H^+H^-$ & $reg^+reg^-$ & $ref~buy^+nb^-buy^0$
		& $\checkmark$ & $\times$ & $\checkmark$ & {} \\
		
		X & $H^+ H^-$ & $reg^+reg^-$ & $ref~nb^+buy^-buy^0$
		& $\checkmark$ & $\times$ & $\times$ & {} \\
		
		\midrule
		\multicolumn{8}{c}{Under the assumptions $ A1', A2'$, and $ A3 $} \\
		\midrule
		
		XI & $H^+L^-$ & $reg^+dis^-$ & $ac~buy^+nb^-buy^0$
		& $\checkmark$ & $\checkmark$ & $\times$ & $\times$ \\
		
		\textcolor{blue}{XII} ($b^{des}$) & \textcolor{blue}{$H^+L^-$} & \textcolor{blue}{$reg^+dis^-$} & \textcolor{blue}{$ac~buy^+nb^-nb^0$}
		& \textcolor{blue}{$\checkmark$} & \textcolor{blue}{$\checkmark$} & \textcolor{blue}{$\checkmark$} & \textcolor{blue}{$\checkmark$} \\
		
		XIII & $L^+ L^-$ & $dis^+dis^-$ & $ref~buy^+buy^-nb^0$
		& $\checkmark$ & $\checkmark$ & $\times$ & $\times$ \\
		
		XIV & $L^+L^-$ & $dis^+dis^-$ & $ref~nb^+buy^-nb^0$
		& $\checkmark$ & $\checkmark$ & $\times$ & $\times$ \\
		
		\textcolor{blue}{XV} & \textcolor{blue}{$L^+ L^-$} & \textcolor{blue}{$dis^+dis^-$} & \textcolor{blue}{$ref~nb^+nb^-nb^0$}
		& \textcolor{blue}{$\checkmark$} & \textcolor{blue}{$\checkmark$} & \textcolor{blue}{$\checkmark$} & \textcolor{blue}{$\checkmark$} \\
		
		\textcolor{blue}{XVI} & \textcolor{blue}{$L^+L^-$} & \textcolor{blue}{$dis^+dis^-$} & \textcolor{blue}{$ref~buy^+nb^-nb^0$}
		& \textcolor{blue}{$\checkmark$} & \textcolor{blue}{$\checkmark$} & \textcolor{blue}{$\checkmark$} & \textcolor{blue}{$\checkmark$} \\
		
		XVII & $H^+H^-$ & $reg^+reg^-$ & $ref~buy^+buy^-nb^0$
		& $\checkmark$ & $\times$ & $\times$ & {} \\
		
		XVIII & $H^+ H^-$ & $reg^+reg^-$ & $ref~nb^+nb^-nb^0$
		& $\checkmark$ & $\times$ & $\checkmark$ & {} \\
		
		XIX & $H^+H^-$ & $reg^+reg^-$ & $ref~buy^+nb^-nb^0$
		& $\checkmark$ & $\times$ & $\times$ & {} \\
		
		XX & $H^+ H^-$ & $reg^+reg^-$ & $ref~nb^+buy^-nb^0$
		& $\checkmark$ & $\times$ & $\checkmark$ & {} \\
		
		\bottomrule
	\end{tabular}
	
	\caption{Sequential equilibria of both models under the assumptions ${\bf A1, A2, A3 }$ or ${\bf A1', A2', A3 }$}
	\label{Seq.eq.:both}
\end{table}
The following two propositions provide the results for Model~1 and
Model~2. These results demonstrate that (a) moving to the sequential
equilibrium dramatically reduces the number of equilibria and (b) the
coordination problem simplifies through the use of BCT, because information
can credibly be shared and actions (payment) are linked with the
signal.\footnote{Technically, we identify actions in Model~1 with actions in
	Model~2 in the canonic way. Therefore, we get a bijection of strategies in
	the models that allows us to compare sets of equilibria.} The Nash and
sequential equilibria in Model~1 and 2 are displayed in
Table~\ref{Seq.eq.:both}, the proofs are relegated to the appendix.
%       \newpage

\begin{prop}[Sequential equilibria  without BCT] \hfill 
	\label{prop:1}
	\begin{enumerate}
		\item Under Assumptions $A1$ and $A2$, there are 5 sequential equilibria in
		pure strategies of the game in Model~1 given by
		$E^1=\{\mbox{I, III, IV, VII, IX}\}$.
		\item Under Assumptions $A1'$ and $A2'$, there are 5 sequential equilibria
		in pure strategies of the game in Model~1 given by
		${E^1}'=\{\mbox{XII, XV, XVI, XVIII, XX}\}$.
		\item Under either set of assumptions, $A1$ and $A2$ or $A1'$ and $A2'$,
		there is a belief system $\beta^{des}$ such that
		$(b^{des},\beta^{des})$ is a sequential equilibrium.
	\end{enumerate}
	
\end{prop}


\begin{prop}[Sequential equilibria with BCT] \hfill 
	\label{prop:2}
	\begin{enumerate}
		\item Under Assumptions $A1$, $A2$, and $A3$, there are 3 sequential equilibria in
		pure strategies of the game in Model~2 given by the
		set $E^2=\{\mbox{I, III, IV}\}$. Further, $E^2
		\subset E^1$.
		
		\item Under Assumptions $A1'$, $A2'$, and $A3$, there are 3 sequential equilibria
		in pure strategies of the game in Model~2 given by the
		set ${E^2}'=\{\mbox{XII, XV, XVI}\}$. Further, ${E^2}' \subset {E^1}'$.
		\item Under either set of assumptions, $A1$, $A2$, and $A3$ or $A1'$,
		$A2'$, and $A3$, there is a belief system $\beta^{des}$ such that
		$(b^{des},\beta^{des})$ is a sequential equilibrium in the game of Model~2.
	\end{enumerate}
\end{prop}


The propositions show in particular that the desired strategy profile
$b^{des}$ is (a part of) a sequential equilibrium in both Model~1 and Model~2. Recall that the two sets of assumptions reflect the cases, in which the
buyer has a positive or negative expected utility from buying the product
without any further knowledge.

A comparison of the propositions shows that the use of BCT in information retrieval shrinks the set of
equilibria. Interestingly, when we take a closer look at which equilibria drop
out, it turns out that using BCT eliminates all equilibria, in which the
expert reports high quality in case that the quality is low (in Model~1 there
are such equilibria; see last two columns of Table~\ref{Seq.eq.:both}). It follows that an (equilibrium) action of paying the
discounted price is an accurate signal for low quality when using BCT. This
observation is independent of the buying incentives of an uninformed buyer.

The number of equilibria can, however, not be reduced to one as already argued above. This
is because the uninformative strategy, in which the expert always pays the discounted price
and the buyer never acquires information is an ``unavoidable'' and equilibrium
strategy. But, the desired equilibrium strategy is payoff dominant
(cf.~Table~\ref{matrix:Model1} and Table~\ref{matrix:Model2}), and this fact
reconciles with the left over coordination problem in Model~2. Thus, also from
this perspective, the desired equilibrium is desired by the two agents as a
group.








\section{Discussion}\label{sec:discussion}
%\section{Conclusion}

Our paper shows that storing quality signals in a blockchain with signal
dependent costs reduces incentives for strategic lying between informed and
uninformed agents, as verifiable payments serve as cost-backed quality
signals. Due to its built-in features of immutability, decentralization,
transparency, and distributed trust, BCT provides a
reliable infrastructure for quality signals.

Our analysis is kept simple and limited to a one-shot interaction between two
players. In other words, our model with anonymous agents does not accommodate
considerations for reputational effects as they were studied in the seminal
paper by \cite{shapiro1982consumer}.  However, future endeavors could involve
devising a repeated game setup wherein our models serve as the foundation for
the stage game. Giving up anonymity of agents, in such a model we could
explore in how far BCT helps to build up (long-term)
reputation.

As an implication, the quality signal can leverage the design of
superior reputation systems that base on BCT, improving
data-sharing platforms \citep{hawlitscheketal} and collaboration platforms
\citep{narang2019design}. Our approach builds on payments as reliable quality
signals, incentivizing buyers to reveal the real quality of a product. When a
secure transaction environment is established, buyers might also sell their
ratings to other market participants or exchange them for other
ratings. Envisioning a scenario that allows buyers and sellers to trade
quality signals \citep{hemmrich2023designing}, our results may have a profound
impact on how companies organize their relationships \citep{becketal}, do
marketing \citep{herhausen2020digital}, select sellers or buyers
\citep{ekstrom2005reputation,mcknight2017distinguishing}, and create social
welfare \citep{arrow1950difficulty}.  Aligning the payoffs with acting
honestly in an open system also allows the creation of trustfully generated
digital information and assets without institutional legitimacy, promoting
digital rating representations of non-digital real-world objects
\citep{pereira2019blockchain}. These quality signals may reduce information
asymmetries and help with selection problems starting with the market for
lemons \citep{akerlof1978market} up to data oracles \citep{caldarelli2020understanding}.







%\bmhead{Acknowledgements}
%
%Acknowledgements are not compulsory. Where included they should be brief. Grant or contribution numbers may be acknowledged.
%
%Please refer to Journal-level guidance for any specific requirements.

\section*{Declarations}

This work was partially supported by the German Research Foundation (DFG) within the Collaborative Research Center 901 ``On-The-Fly Computing'' (SFB 901) under the project number 160364472-SFB901. Papatya Duman was partially supported by the German Research Foundation (DFG) under the project number 542389523. The authors have nothing further to disclose. 




%%===================================================%%
%% For presentation purpose, we have included        %%
%% \bigskip command. Please ignore this.             %%
%%===================================================%%


\begin{appendices}

\section{Determination of  Nash and sequential equilibria under ${\bf A1 }$ - ${\bf A3 }$}
\label{sec:seq}

\ref{sec:seq}  provides the calculations and hence proofs of
Propositions~\ref{prop:1} and \ref{prop:2} under the assumptions $ A1 $, $ A2 $,
and, whenever applicable, $ A3 $. In \ref{sec:eq.comp:Model12} %\ref{sec:neg.exp.uti.} 
we display the same
analysis under assumptions $ A1', A2', $ and $ A3 $. Finally  \ref{sec:zero.exp.ut.} treats the case
\mbox{$uq-p=0$}, in which a completely uninformed buyer is indifferent between buying
and not buying the product.

\medskip


{\noindent\bf Nash equilibria in the games of Models~1 and 2}\\
Model~1 is informationally equivalent to the simultaneous move game in which
the expert has the 4 strategies $(H^+H^-), (H^+L^-), (L^+H^-), (L^+L^-)$ and the
buyer's strategy set $$ \left\{ ac, ref\right\} \times \left\{ buy^+, nb^+ \right\} \times \left\{ buy^-, nb^- \right\} \times \left\{ buy^0, nb^0 \right\}$$ has 16 members as he has 2 possible
actions at each of his four information sets. Model~1's pure strategy Nash
equilibria are found via the normal-form representation of that simultaneous
move game. Those equilibria are readily determined by payoff comparisons and
are shown in Table~\ref{matrix:Model1}. We reach the Nash equilibria in the game in  Model~2 via similar
argumentation (see Table~\ref{matrix:Model2}). Here the expert's strategies
are $ (reg^+reg^-), (reg^+dis^-), (dis^+reg^-), (dis^+dis^-)$.
The game in Model~1 has 10 Nash equilibria in pure strategies,
while the game in Model~2 has 6 Nash equilibria.

\medskip

{\noindent\bf Bayesian Consistency}\\
Next, we will search for the sequential ones among those equilibria.  We
denote the expert and the buyer as Player 1 and~2, respectively. Let
$ b_1,b_2 $ be their completely mixed behavioral strategies. The corresponding
Bayesian consistent beliefs will be computed as follows:
$ \beta^1 \in \mathbb{R}^4$ represents the beliefs of Player 2 at the first
information set from left (orange). $ \beta^1_1$ is the belief that the
decision node at the bottom is reached, i.e., the product quality is high and
Player 1 has played $ H^+ $, while $ \beta^1_2$ is the belief that the second
decision node from bottom is reached, i.e., the product quality is high and
Player 1 has played $ L^+ $. In case of the low quality product, Player 2
believes that Player 1 has played $ H^-, L^- $ with probability
$ \beta^1_3, \beta^1_4$, respectively. Under the assumption that $ b_1,b_2 $
are completely mixed, the belief $ \beta^1 $ can be computed via conditional
probability:
%\begin{align}
%	\beta^1 =\left(\beta^1_1,\beta^1_2,\beta^1_3,\beta^1_4\right)= (sq, (1-s)q, t(1-q), (1-q) (1-t))
%\end{align}

\begin{align}
	\beta^1 =(\beta^1_1,\beta^1_2,\beta^1_3,\beta^1_4)= (sq, (1-s)q, t(1-q), (1-q) (1-t))
\end{align}
%$$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\beta^1 =(\beta^1_1,\beta^1_2,\beta^1_3,\beta^1_4)= (sq, (1-s)q, t(1-q), (1-q) (1-t)) \qquad~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ (A.1)$$
where $ s=b_1(H^+), t=b_1(H^-) $. It is clear by construction that $ \beta^1_1+\beta^1_2=q, \beta^1_3+\beta^1_4=1-q $.

$ \beta^2 \in \mathbb{R}^2$ represents the beliefs of the buyer at the second information set from left (green). When this information set is reached, Player 2 believes with probability $ \beta^2_1 $ that the decision node at the bottom is reached. Similarly, the belief $ \beta^2 $ can be computed via conditional probability:
%\begin{align}
%	\beta^2 =\left( \beta^2_1,\beta^2_2\right)= \left(\frac{sq}{sq + t(1-q)}, \frac{t (1-q)}{sq + t(1-q)}\right).
%\end{align}

\begin{align}
	\beta^2 =\left( \beta^2_1,\beta^2_2\right)= \left(\frac{sq}{sq + t(1-q)}, \frac{t (1-q)}{sq + t(1-q)}\right).
\end{align}

%$$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~	\beta^2 =\left( \beta^2_1,\beta^2_2\right)= \left(\frac{sq}{sq + t(1-q)}, \frac{t (1-q)}{sq + t(1-q)}\right). \qquad ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(A.2)$$
$ \beta^3 \in \mathbb{R}^2$ represents the beliefs of the buyer at the third information set from left (pink). Via similar argumentation, one can see that  
%\begin{align}
%	\beta^3 =\left(\beta^3_1,\beta^3_2\right)= \left(\frac{q (1-s)}{q (1-s) + (1-q)(1-t)}, \frac{(1-q)(1-t)}{q (1-s)+ (1-q)(1-t)}\right).
%\end{align}

\begin{align}\beta^3 =\left(\beta^3_1,\beta^3_2\right)= \left(\frac{q (1-s)}{q (1-s) + (1-q)(1-t)}, \frac{(1-q)(1-t)}{q (1-s)+ (1-q)(1-t)}\right).	
\end{align}
%	$$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\beta^3 =\left(\beta^3_1,\beta^3_2\right)= \left(\frac{q (1-s)}{q (1-s) + (1-q)(1-t)}, \frac{(1-q)(1-t)}{q (1-s)+ (1-q)(1-t)}\right). \qquad ~~~~~~~~~~~(A.3)$$
The fourth information set of Player 2 is represented in blue. The belief vector of Player 2 for this information set is $ \beta^4 $, and the beliefs in reaching decision nodes from bottom to top are as follows:  
%\begin{align}
%	\beta^4 =\left(\beta^4_1,\beta^4_2,\beta^4_3,\beta^4_4\right) = (sq, (1-s)q, t (1-q), (1-t)(1-q)).
%\end{align}

\begin{align}
	\beta^4 =\left(\beta^4_1,\beta^4_2,\beta^4_3,\beta^4_4\right) = (sq, (1-s)q, t (1-q), (1-t)(1-q)).
\end{align}
%$$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\beta^4 =\left(\beta^4_1,\beta^4_2,\beta^4_3,\beta^4_4\right) = (sq, (1-s)q, t (1-q), (1-t)(1-q)). \qquad ~~~~~~~~~~~~~~(A.4)$$

	\begin{sidewaystable}[htbp]
	%	\centering 
	%\vspace*{15cm}
	\resizebox{\textwidth}{!}{%
		\begin{tabular}{cl|cc||cc||cc|cc|}
			& \multicolumn{1}{c}{} & \multicolumn{8}{c}{Expert} \\
			& \multicolumn{1}{c}{} & \multicolumn{8}{c}{(Player 1)} \\
			& \multicolumn{1}{c}{} & \multicolumn{2}{c}{$H^+H^-$}  & \multicolumn{2}{c}{$H^+L^-$}  & \multicolumn{2}{c}{$L^+H^-$} & \multicolumn{2}{c}{$L^+L^-$}  \\ 
			\cmidrule{3-10}& $ac~	buy^+buy^-buy^0$ & $ uq-p-x $& $(v+r)q-p+x$ & $uq-p-x$ &\colorbox{yellow}{$ vq-p+x+r $}&$ uq-p-x$&$ vq-p+x $ &$ uq-p-x$ & $ (v+r)q-p+x $\\ 
			\cmidrule{3-10}& $ac~buy^+nb^-buy^0$ & $uq-p-x$ & \colorbox{yellow}{$(v+r)q-p+x$} & \colorbox{yellow}{$(u-p)q-x$} &\colorbox{yellow}{$ (v+r)q-p+x $}&$ pq-p-x$&$ vq-p+x $&$ -x $&$ vq-p+x $\\ 
			\cmidrule{3-10}& $ac~buy^+buy^-nb^0$ & $uq-p-x$ & $(v+r)q-p+x$ & $uq-p-x
			$ &\colorbox{yellow}{$ vq-p+x+r $}&$ uq-p-x $&$ vq-p+x $&$ uq-p-x$ & $ (v+r)q-p+x $
			\\\cmidrule{3-10}
			& $ac~buy^+nb^-nb^0$ & $uq-p-x$ & \colorbox{yellow}{$(v+r)q-p+x$} & \colorbox{yellow}{$(u-p)q-x$} &\colorbox{yellow}{$ (v+r)q-p+x $}&$ pq-p-x$&$ vq-p+x $&$ -x $&$ vq-p+x $\\\cmidrule{3-10}
			& $ac~nb^+buy^-buy^0$ & $-x$ & $vq-p+x$ & $pq-p-x$ &\colorbox{yellow}{$ (v+r)q-p+x $}&\colorbox{yellow}{$ (u-p)q-x$}&$ vq-p+x $&$ uq-p-x$ & \colorbox{yellow}{$ (v+r)q-p+x $}\\ \cmidrule{3-10}
			& $ac~nb^+nb^-buy^0$ & $-x$ & \colorbox{yellow}{$vq-p+x$} & $-x$ &\colorbox{yellow}{$vq-p+x$}&$-x$&\colorbox{yellow}{$vq-p+x$}&$ -x $&\colorbox{yellow}{$vq-p+x$}\\\cmidrule{3-10}
			& $ac~nb^+buy^-nb^0$ & $-x$ & $vq-p+x$ & $pq-p-x$ &\colorbox{yellow}{$ (v+r)q-p+x $}&\colorbox{yellow}{$ (u-p)q-x$}&$ vq-p+x $&$ uq-p-x$ & \colorbox{yellow}{$ (v+r)q-p+x $}\\\cmidrule{3-10}
			Buyer & $ac~nb^+nb^-nb^0$ & $-x$ & \colorbox{yellow}{$vq-p+x$} & $-x$ &\colorbox{yellow}{$vq-p+x$}&$-x$&\colorbox{yellow}{$vq-p+x$}&$ -x $&\colorbox{yellow}{$vq-p+x$}\\\cmidrule{3-10}
			(Player 2)& $ref~buy^+buy^-buy^0$ & \colorbox{yellow}{$uq-p$} & \colorbox{yellow}{$vq-p$} & $uq-p$ & \colorbox{yellow}{$vq-p$}&$uq-p$ & \colorbox{yellow}{$vq-p$}&\colorbox{yellow}{$uq-p$} & \colorbox{yellow}{$vq-p$}\\ \cmidrule{3-10}
			& $ref~buy^+nb^-buy^0$ &  \colorbox{yellow}{$uq-p$} & \colorbox{yellow}{$vq-p$} & $uq-p$ & \colorbox{yellow}{$vq-p$}&$uq-p$ & \colorbox{yellow}{$vq-p$}&\colorbox{yellow}{$uq-p$} & \colorbox{yellow}{$vq-p$}\\\cmidrule{3-10}
			& $ref~buy^+buy^-nb^0$ & $0$ & \colorbox{yellow}{$vq-p$} & $0$ & \colorbox{yellow}{$vq-p$}&$0$ & \colorbox{yellow}{$vq-p$}&$0$ & \colorbox{yellow}{$vq-p$}\\\cmidrule{3-10}
			& $ref~buy^+nb^-nb^0$ & $0$ & \colorbox{yellow}{$vq-p$} & $0$ & \colorbox{yellow}{$vq-p$}&$0$ & \colorbox{yellow}{$vq-p$}&$0$ & \colorbox{yellow}{$vq-p$}\\\cmidrule{3-10}
			& $ref~nb^+buy^-buy^0$ &  \colorbox{yellow}{$uq-p$} & \colorbox{yellow}{$vq-p$} & $uq-p$ & \colorbox{yellow}{$vq-p$}&$uq-p$ & \colorbox{yellow}{$vq-p$}&\colorbox{yellow}{$uq-p$} & \colorbox{yellow}{$vq-p$}\\ \cmidrule{3-10}
			& $ref~nb^+nb^-buy^0$ &  \colorbox{yellow}{$uq-p$} & \colorbox{yellow}{$vq-p$} & $uq-p$ & \colorbox{yellow}{$vq-p$}&$uq-p$ & \colorbox{yellow}{$vq-p$}&\colorbox{yellow}{$uq-p$} & \colorbox{yellow}{$vq-p$}\\\cmidrule{3-10}
			& $ref~nb^+buy^-nb^0$ & $0$ & \colorbox{yellow}{$vq-p$} & $0$ & \colorbox{yellow}{$vq-p$}&$0$ & \colorbox{yellow}{$vq-p$}&$0$ & \colorbox{yellow}{$vq-p$}\\\cmidrule{3-10}
			& $ref~nb^+nb^-nb^0$ & $0$ & \colorbox{yellow}{$vq-p$} & $0$ & \colorbox{yellow}{$vq-p$}&$0$ & \colorbox{yellow}{$vq-p$}&$0$ & \colorbox{yellow}{$vq-p$}\\\cmidrule{3-10}
		\end{tabular}
	}
	\caption{The normal-form representation of Model~1 in which the
		players' best responses are marked.}
	\label{matrix:Model1}
\end{sidewaystable} 
	


\begin{sidewaystable}[htbp]
	%	\centering
	%\vspace*{15cm}
	\resizebox{\textwidth}{!}{%
		\begin{tabular}{cl|cc||cc||cc|cc|}
			& \multicolumn{1}{c}{} & \multicolumn{8}{c}{Expert} \\
			& \multicolumn{1}{c}{} & \multicolumn{8}{c}{(Player 1)} \\
			& \multicolumn{1}{c}{} & \multicolumn{2}{c}{$reg^+reg^-$}  & \multicolumn{2}{c}{$reg^+dis^-$}  & \multicolumn{2}{c}{$dis^+reg^-$} & \multicolumn{2}{c}{$dis^+dis^-$}\\\cmidrule{3-10}
			& $ac~	buy^+buy^-buy^0$ & $ uq-p-x $& $(v+r)q-p+x$ & $uq-p-x$ & \colorbox{yellow}{$(v-p+\pbar)q-\pbar+x+r$}&$ uq-p-x$&$ (v+p-\pbar)q-p+x
			$ &$ uq-p-x$ & $ (v-r)q-\pbar+x+r $\\ \cmidrule{3-10} 
			& $ac~buy^+nb^-buy^0$ & $uq-p-x$ & $(v+r)q-p+x$ & \colorbox{yellow}{$(u-p)q-x$} &\colorbox{yellow}{$ (v-p+\pbar+r)q-\pbar+x
				$}&$ pq-p-x$&$ (v+p-\pbar)q-p+x$&$ -x $&$ vq-\pbar+x$\\\cmidrule{3-10} 
			& $ac~buy^+buy^-nb^0$ & $uq-p-x$ & $(v+r)q-p+x$ & $uq-p-x
			$ &\colorbox{yellow}{$ (v-p+\pbar)q-\pbar+x+r $}&$ uq-p-x $&$ (v+p-\pbar)q-p+x
			$&$ uq-p-x$ & $ (v-r)q-\pbar+x+r $
			\\\cmidrule{3-10}
			& $ac~buy^+nb^-nb^0$ & $uq-p-x$ & $(v+r)q-p+x$ & \colorbox{yellow}{$(u-p)q-x$} &\colorbox{yellow}{$ (v-p+\pbar+r)q-\pbar+x$}&$ pq-p-x$&$ (v+p-\pbar)q-p+x
			$&$ -x $&$ vq-\pbar+x$\\\cmidrule{3-10}
			& $ac~nb^+buy^-buy^0$ & $-x$ & $vq-p+x$ & $pq-p-x$ &$ (v-p+\pbar-r)q-\pbar+x+r$&\colorbox{yellow}{$ (u-p)q-x$}&$ (v+p-\pbar)q-p+x
			$&$ uq-p-x$ & \colorbox{yellow}{$ (v-r)q-\pbar+x+r $}\\ \cmidrule{3-10}
			& $ac~nb^+nb^-buy^0$ & $-x$ & $vq-p+x$ & $-x$ &$(v-p+\pbar)q-\pbar+x$&$-x$&$ (v+p-\pbar)q-p+x
			$&$ -x $&\colorbox{yellow}{$ vq-\pbar+x$}\\\cmidrule{3-10}
			& $ac~nb^+buy^-nb^0$ & $-x$ & $vq-p+x$ & $pq-p-x$ &$ (v-p+\pbar-r)q-\pbar+x+r$&\colorbox{yellow}{$ (u-p)q-x$}&$ (v+p-\pbar)q-p+x
			$&$ uq-p-x$ & \colorbox{yellow}{$ (v-r)q-\pbar+x+r  $}\\\cmidrule{3-10}
			Buyer& $ac~nb^+nb^-nb^0$ & $-x$ & $vq-p+x$ & $-x$ &$(v-p+\pbar)q-\pbar+x$&$-x$&$ (v+p-\pbar)q-p+x
			$&$ -x $&\colorbox{yellow}{$ vq-\pbar+x$}\\\cmidrule{3-10}
			(Player 2)& $ref~buy^+buy^-buy^0$ & \colorbox{yellow}{$uq-p$} & $vq-p$ & $uq-p$ & $q(v-p+\pbar)-\pbar$&$uq-p$ & $q(v+p-\pbar)-p
			$&\colorbox{yellow}{$uq-p$} & \colorbox{yellow}{$vq-p$}\\ \cmidrule{3-10}
			& $ref~buy^+nb^-buy^0$ &  \colorbox{yellow}{$uq-p$} & $vq-p$ & $uq-p$ & $q(v-p+\pbar)-\pbar$&$uq-p$ & $q(v+p-\pbar)-p
			$&\colorbox{yellow}{$uq-p$} & \colorbox{yellow}{$vq-p$}\\\cmidrule{3-10}
			& $ref~buy^+buy^-nb^0$ & $0$ & $vq-p$ & $0$ & $q(v-p+\pbar)-\pbar$&$0$ & $q(v+p-\pbar)-p
			$&$0$ & \colorbox{yellow}{$vq-p$}\\\cmidrule{3-10}
			& $ref~buy^+nb^-nb^0$ & $0$ & $vq-p$ & $0$ & $q(v-p+\pbar)-\pbar$&$0$ & $q(v+p-\pbar)-p
			$&$0$ & \colorbox{yellow}{$vq-p$}\\\cmidrule{3-10}
			& $ref~nb^+buy^-buy^0$ &  \colorbox{yellow}{$uq-p$} & $vq-p$ & $uq-p$ & $q(v-p+\pbar)-\pbar$&$uq-p$ & $q(v+p-\pbar)-p
			$&\colorbox{yellow}{$uq-p$} & \colorbox{yellow}{$vq-p$}\\ \cmidrule{3-10}
			& $ref~nb^+nb^-buy^0$ &  \colorbox{yellow}{$uq-p$} & $vq-p$ & $uq-p$ & $q(v-p+\pbar)-\pbar$&$uq-p$ & $q(v+p-\pbar)-p
			$&\colorbox{yellow}{$uq-p$} & \colorbox{yellow}{$vq-p$}\\\cmidrule{3-10}
			& $ref~nb^+buy^-nb^0$ & $0$ & $vq-p$ & $0$ & $q(v-p+\pbar)-\pbar$&$0$ & $q(v+p-\pbar)-p
			$&$0$ & \colorbox{yellow}{$vq-p$}\\\cmidrule{3-10}
			& $ref~nb^+nb^-nb^0$ & $0$ & $vq-p$ & $0$ & $q(v-p+\pbar)-\pbar$&$0$ & $q(v+p-\pbar)-p
			$&$0$ & \colorbox{yellow}{$vq-p$}\\\cmidrule{3-10}
	\end{tabular}
}
	\caption{The normal-form representation of Model~2 in which the
		players' best responses are marked.}
	\label{matrix:Model2}
\end{sidewaystable} 







{\noindent\bf Sequential equilibria in the game of Model~1}\\
For any Nash equilibria in Model 1, one can easily construct the equilibrium
strategy as a behavioral strategy. Then, that behavioral strategy $ b $ along
with a belief system $ \beta $ must satisfy Consistency and Sequential
Rationality. To do so, for any $ b $, we must find a $ \beta $, so that
together they satisfy Sequential Rationality. Then, a sequence of assessments
$ (b^m, \beta^m)_{m \in \mathbb{N}} $ of completely mixed $ b^m $, Bayesian
consistent $ \beta^m $ converging to $ (b, \beta) $ is necessary. If an
equilibrium is not sequential, it will be justified here why this is the case. 
Below in Table~\ref{Seq.eq.:Model1}, one can see that 5 out of 10 Nash equilibria in
Model 1 are sequential equilibria. The corresponding belief systems and
the assessment sequences that allow us to confirm sequential rationality and
consistency are shown in Table~\ref{table:assessmod1}.


\begin{table}[h]
	\centering
	\begin{tabular}{ |p{1cm}||p{2cm}|p{3cm}|p{4cm}|  }
		\hline
		\multicolumn{4}{|c|}{NE in Model 1 under $ A1 $-$ A2 $} \\ 
		\hline
		& Player 1 & Player 2 & Seq. Equilibrium\\
		\hline
		I   & $H^+L^-$    &$ac~buy^+nb^-buy^0$&  $ \checkmark $ \\
		II&  $H^+L^-$ & $ac~buy^+nb^-nb^0$   & $ \times $\\
		III &$L^+ L^-$ & $ref~buy^+buy^-buy^0$&  $ \checkmark $\\
		IV    &$L^+L^-$ & $ref~nb^+buy^-buy^0$&  $ \checkmark $\\
		V&   $L^+ L^-$  & $ref~nb^+nb^-buy^0$&$ \times $\\
		VI& $L^+L^-$  & $ref~buy^+nb^-buy^0$   &$ \times $\\
		VII& $H^+H^-$  & $ref~buy^+buy^-buy^0$&$ \checkmark $\\
		VIII& $H^+ H^-$  & $ref~nb^+nb^-buy^0$&$ \times $\\
		IX& $H^+H^-$  & $ref~buy^+nb^-buy^0$&$ \checkmark $\\
		X& $H^+ H^-$  & $ref~nb^+buy^-buy^0$&$ \times $\\
		\hline
	\end{tabular} 
	\caption{Sequential equilibria in Model~1 under ${\bf A1 }$-${\bf A2 }$}
	\label{Seq.eq.:Model1}
\end{table}

It remains to show that the other Nash equilibria are not sequential
equilibria. We show this separately.

\begin{itemize}
	\item \textbf{Equilibrium II:} For any $ b^m_1(H^+), b^m_1(H^-) \in (0, 1)$, $ \mathbb{E}_2[\mbox{buy}^0] > \mathbb{E}_2[\mbox{nb}^0]$ as 
	$ \mathbb{E}_2[\mbox{buy}^0]=(u-p)\beta^4_1+(u-p)\beta^4_2+ (-p)\beta^4_3+(-p)\beta^4_4=(u-p)q-p(1-q)=uq-p > 0=\mathbb{E}_2[\mbox{nb}^0].$
	By sequential rationality, this equilibrium cannot be a sequential equilibrium.
	\item \textbf{Equilibrium V and VI:} For any completely mixed strategies $ b^m $ with $s_m=b^m_1 (H^+) \rightarrow 0, t_m=b^m_1 (H^-)\rightarrow 0$, consider the corresponding Bayesian consistent belief at the third information set as in (A.3): $ \beta^{m3} \longrightarrow (q,1-q)$.
	By sequential rationality, Player 2 plays $ buy^-$ at his third information set as $ \mathbb{E}_2[\mbox{buy}^-]=(u-p-x)q+(-p-x)(1-q)= uq-p-x >-x=\mathbb{E}_2[\mbox{nb}^-] $.   
	\item \textbf{Equilibrium VIII and X:} For any completely mixed strategies $ b^m $ with $s_m=b^m_1 (H^+) \rightarrow 1, t_m=b^m_1 (H^-)\rightarrow 1$, consider the corresponding Bayesian consistent belief at the second information set as in (A.2):
	$ 	\beta^{m2} \longrightarrow (q, 1-q) $.
	By sequential rationality, Player 2 plays $ buy^+ $ at his second information set as $ \mathbb{E}_2[\mbox{buy}^+]=(u-p-x)q+(-p-x)(1-q)= uq-p-x >-x=\mathbb{E}_2[\mbox{nb}^+] $.
\end{itemize}


\begin{table}
	\centering
	\begin{tabular}{ | c || l | p{7cm} |}
		\hline
		Equilibrium & Belief System & Assessment Sequence ($ m\geq2 $) \\ \hline
		I & \begin{tabular}{ l }
			\\
			$ \beta^1=(q,0,0,1-q) $  \\ 
			$ \beta^2=(1,0)$   \\ 
			$ \beta^3=(0,1) $   \\ 
			$ \beta^4=(q,0,0,1-q) $ 
			
		\end{tabular} & \begin{tabular}{ l }
			
			$ b^m_1(H^+)=1-1/m, b^m_1(H^-)=1/m $  \\ 
			$ b^m_2(ac)=1-1/m, b^m_2(buy^-)=1/m$   \\ 
			$ b^m_2(buy^+)=b^m_2(buy^0)=1-1/m $   \\ 
			
		\end{tabular}  \\ \hline
		III & \begin{tabular}{ l }
			\\
			$ \beta^1=(0,q,0,1-q) $  \\ 
			$ \beta^2=(q,1-q)$   \\ 
			$ \beta^3=(q,1-q) $   \\ 
			$ \beta^4=(0,q,0,1-q) $ 
			
		\end{tabular} & \begin{tabular}{ l }
			
			$ b^m_1(H^+)=b^m_1(H^-)=1/m $  \\ 
			$ b^m_2(ac)=1/m, b^m_2(buy^+)=1-1/m$   \\ 
			$ b^m_2(buy^-)=b^m_2(buy^0)=1-1/m $   \\ 
			
		\end{tabular}  \\ \hline
		IV & \begin{tabular}{ l }
			\\
			$ \beta^1=(0,q,0,1-q) $  \\ 
			$ \beta^2=(0,1)$   \\ 
			$ \beta^3=(q,1-q) $   \\ 
			$ \beta^4=(0,q,0,1-q) $ 
			
		\end{tabular} & \begin{tabular}{ l }
			
			$ b^m_1(H^+)=1/m^2, b^m_1(H^-)=1/m $  \\ 
			$ b^m_2(ac)=b^m_2(buy^+)=1/m$   \\ 
			$ b^m_2(buy^-)=b^m_2(buy^0)=1-1/m $   \\ 
			
		\end{tabular}  \\\hline
		VII & \begin{tabular}{ l }
			\\
			$ \beta^1=(q,0,1-q,0) $  \\ 
			$ \beta^2=(q,1-q)$   \\ 
			$ \beta^3=(q,1-q) $   \\ 
			$ \beta^4=(q,0,1-q,0) $ 
			
		\end{tabular} & \begin{tabular}{ l }
			
			$ b^m_1(H^+)=b^m_1(H^-)=1-1/m $  \\ 
			$ b^m_2(ac)=1/m, b^m_2(buy^+)=1-1/m$   \\ 
			$ b^m_2(buy^-)=b^m_2(buy^0)=1-1/m $   \\ 
			
		\end{tabular}  \\\hline
		IX & \begin{tabular}{ l }
			\\
			$ \beta^1=(q,0,1-q,0) $  \\ 
			$ \beta^2=(q,1-q)$   \\ 
			$ \beta^3=(0,1) $   \\ 
			$ \beta^4=(q,0,1-q,0) $ 
			
		\end{tabular} & \begin{tabular}{ l }
			
			$ b^m_1(H^+)=1-1/m^2, b^m_1(H^-)=1-1/m $  \\ 
			$ b^m_2(ac)=1/m, b^m_2(buy^-)=1/m$   \\ 
			$ b^m_2(buy^+)=b^m_2(buy^0)=1-1/m $   \\ 
			
		\end{tabular}  \\
		\hline
	\end{tabular}
	\caption{Belief systems for equilibria in Model~1 under ${\bf A1 }$-${\bf A2 }$: For the sequence of
		assessments, the corresponding Bayesian consistent beliefs are
		calculated as in (A.1)-(A.4).}
	\label{table:assessmod1}
\end{table}

{\noindent\bf Sequential equilibria in the game of Model~2}\\
Consider again a completely mixed behavioral strategy $(b_1, b_2)$ and the
Bayesian consistent belief system $ \beta $ as described above. In
Tables~\ref{Seq.eq.:Model2} and \ref{table:assessmod2} we see that 3 out of 6
Nash equilibria are sequential and the corresponding belief
systems.
We remark that Equilibrium I shows the desired behavioral strategy and that
the set of sequential equilibria in Model~2 is included in the set of those in Model~1.


\begin{table}[h]
	\centering
	\begin{tabular}{ |p{1cm}||p{3cm}|p{3cm}|p{1cm}|  }
		\hline
		\multicolumn{4}{|c|}{NE in Model 2 under $ A1 $-$ A3 $} \\ 
		\hline
		& Player 1 & Player 2 & SE\\
		\hline
		I   & $reg^+dis^-$    &$ac~buy^+nb^-buy^0$&  $ \checkmark $ \\
		II&  $reg^+dis^-$ & $ac~buy^+nb^-nb^0$   & $ \times $\\
		III &$dis^+dis^-$ & $ref~buy^+buy^-buy^0$&  $ \checkmark $\\
		IV    &$dis^+dis^-$ & $ref~nb^+buy^-buy^0$&  $ \checkmark $\\
		V&   $dis^+dis^-$  & $ref~nb^+nb^-buy^0$&$ \times $\\
		VI& $dis^+dis^-$  & $ref~buy^+nb^-buy^0$   &$ \times $\\
		\hline
	\end{tabular} 
	\caption{Sequential equilibria in Model~2 under ${\bf A1 }$-${\bf A3 }$}
	\label{Seq.eq.:Model2}
\end{table}

\begin{table}[h]
	\centering
	\begin{tabular}{ | c || l | p{7cm} |}
		\hline
		Equilibrium & Belief System & Assessment Sequence ($ m\geq2 $) \\ \hline
		I & \begin{tabular}{ l }
			\\
			$ \beta^1=(q,0,0,1-q) $  \\ 
			$ \beta^2=(1,0)$   \\ 
			$ \beta^3=(0,1) $   \\ 
			$ \beta^4=(q,0,0,1-q) $ 
			
		\end{tabular} & \begin{tabular}{ l }
			
			$ b^m_1(reg^+)=1-1/m, b^m_1(reg^-)=1/m $  \\ 
			$ b^m_2(ac)=1-1/m, b^m_2(buy^-)=1/m$   \\ 
			$ b^m_2(buy^+)=b^m_2(buy^0)=1-1/m $   \\ 
			
		\end{tabular}  \\ \hline
		III & \begin{tabular}{ l }
			\\
			$ \beta^1=(0,q,0,1-q) $  \\ 
			$ \beta^2=(q,1-q)$   \\ 
			$ \beta^3=(q,1-q) $   \\ 
			$ \beta^4=(0,q,0,1-q) $ 
			
		\end{tabular} & \begin{tabular}{ l }
			
			$b^m_1(reg^+)=b^m_1(reg^-)=1/m $  \\ 
			$ b^m_2(ac)=1/m, b^m_2(buy^+)=1-1/m$   \\ 
			$ b^m_2(buy^-)=b^m_2(buy^0)=1-1/m $   \\ 
			
		\end{tabular}  \\ \hline
		IV & \begin{tabular}{ l }
			\\
			$ \beta^1=(0,q,0,1-q) $  \\ 
			$ \beta^2=(0,1)$   \\ 
			$ \beta^3=(q,1-q) $   \\ 
			$ \beta^4=(0,q,0,1-q) $ 
			
		\end{tabular} & \begin{tabular}{ l }
			
			$ b^m_1(reg^+)=1/m^2, b^m_1(reg^-)=1/m $  \\ 
			$ b^m_2(ac)=b^m_2(buy^+)=1/m$   \\ 
			$ b^m_2(buy^-)=b^m_2(buy^0)=1-1/m $   \\ 
			
		\end{tabular}  \\\hline
		
		\hline
	\end{tabular}
	\caption{Belief systems for equilibria in Model~2 under ${\bf A1 }$-${\bf A3 }$: For the sequence of
		assessments, the corresponding Bayesian consistent beliefs are
		calculated as in (A.1)-(A.4).}
	\label{table:assessmod2}
\end{table}

The reasons behind the failure of the other equilibria are as follows:
\begin{itemize}
	\item \textbf{Equilibrium II:} For any $ b^m_1(reg^+), b^m_1(reg^-) \in (0, 1)$, $ \mathbb{E}_2[\mbox{buy}^0] > \mathbb{E}_2[\mbox{nb}^0]$ as 
	$ \mathbb{E}_2[\mbox{buy}^0]=(u-p)\beta^4_1+(u-p)\beta^4_2+ (-p)\beta^4_3+(-p)\beta^4_4=(u-p)q-p(1-q)=uq-p > 0=\mathbb{E}_2[\mbox{nb}^0].$
	By sequential rationality, this equilibrium cannot be a sequential equilibrium.
	\item \textbf{Equilibrium V and VI:} For any completely mixed strategies $ b^m $ with $s_m=b^m_1 (reg^+) \rightarrow 0, t_m=b^m_1(reg^-)\rightarrow 0$, consider the corresponding Bayesian consistent belief at the third information set as in (A.3): $ \beta^{m3} \longrightarrow (q,1-q)$.
	By sequential rationality, Player 2 plays $ buy^-$ at his third information set as $ \mathbb{E}_2[\mbox{buy}^-]=(u-p-x)q+(-p-x)(1-q)= uq-p-x >-x=\mathbb{E}_2[\mbox{nb}^-] $.   
	
\end{itemize}





\section{Determination of  Nash and sequential equilibria under ${\bf A1 ', A2', }$
	and ${\bf A3 }$}
\label{sec:eq.comp:Model12}

We proceed as in the previous section and give the distinction between Nash
and sequential equilibria in Model~1 (Model~2) in Table~\ref{Seq.eq.:Model1B}  (Table~\ref{Seq.eq.:Model2B}) as well as
the corresponding belief systems in Tables~\ref{table:assessmod1B} and \ref{table:assessmod2B} followed by the
contradictions that the remaining Nash equilibria are not sequential.

Again, the set of equilibria shrinks when moving from Model~1 to
Model~2. Under the assumptions the desired strategy here requires the buyer
not to buy the product when uninformed. It is part of the sequential
equilibrium II and prevails in both models.




\begin{table}[h]
	\centering
	\begin{tabular}{ |p{1cm}||p{3cm}|p{3cm}|p{1cm}|  }
		\hline
		\multicolumn{4}{|c|}{NE in Model 1 under $ A1' $, and $ A2' $} \\ 
		\hline
		& Player 1 & Player 2 & SE\\
		\hline
		I   & $H^+L^-$    &$ac~buy^+nb^-buy^0$&  $ \times $ \\
		II&  $H^+L^-$ & $ac~buy^+nb^-nb^0$   & $ \checkmark $\\
		III &$L^+ L^-$ & $ref~buy^+buy^-nb^0$&  $ \times $\\
		IV    &$L^+L^-$ & $ref~nb^+buy^-nb^0$&  $ \times $\\
		V&   $L^+ L^-$  & $ref~nb^+nb^-nb^0$&$ \checkmark $\\
		VI& $L^+L^-$  & $ref~buy^+nb^-nb^0$   &$ \checkmark $\\
		VII& $H^+H^-$  & $ref~buy^+buy^-nb^0$&$ \times $\\
		VIII& $H^+ H^-$  & $ref~nb^+nb^-nb^0$&$ \checkmark $\\
		IX& $H^+H^-$  & $ref~buy^+nb^-nb^0$&$ \times $\\
		X& $H^+ H^-$  & $ref~nb^+buy^-nb^0$&$ \checkmark $\\
		\hline
	\end{tabular} 
	\caption{Sequential equilibria in Model~1 under the assumptions ${\bf A1' }$, and ${\bf A2' }$: From Table~\ref{matrix:Model1}, it is easy to see that 
		there are 10 Nash equilibria in pure strategies under the assumptions.}
	\label{Seq.eq.:Model1B}
\end{table}

\begin{table}[h]
	\centering
	\begin{tabular}{ | c || l | p{7cm} |}
		\hline
		Equilibrium & Belief System & Assessment Sequence ($ m\geq2 $) \\ \hline
		II & \begin{tabular}{ l }
			\\
			$ \beta^1=(q,0,0,1-q) $  \\ 
			$ \beta^2=(1,0)$   \\ 
			$ \beta^3=(0,1) $   \\ 
			$ \beta^4=(q,0,0,1-q) $ 
			
		\end{tabular} & \begin{tabular}{ l }
			
			$ b^m_1(H^+)=1-1/m, b^m_1(H^-)=1/m $  \\ 
			$ b^m_2(ac)=b^m_2(buy^+)=1-1/m$   \\ 
			$ b^m_2(buy^-)=b^m_2(buy^0)=1/m $   \\ 
			
		\end{tabular}  \\ \hline
		V & \begin{tabular}{ l }
			\\
			$ \beta^1=(0,q,0,1-q) $  \\ 
			$ \beta^2=(q,1-q)$   \\ 
			$ \beta^3=(q,1-q) $   \\ 
			$ \beta^4=(0,q,0,1-q) $ 
			
		\end{tabular} & \begin{tabular}{ l }
			
			$ b^m_1(H^+)=b^m_1(H^-)=1/m $  \\ 
			$ b^m_2(ac)=b^m_2(buy^+)=1/m$   \\ 
			$ b^m_2(buy^-)=b^m_2(buy^0)=1/m $   \\ 
			
		\end{tabular}  \\ \hline
		VI & \begin{tabular}{ l }
			\\
			$ \beta^1=(0,q,0,1-q) $  \\ 
			$ \beta^2=(1,0)$   \\ 
			$ \beta^3=(q,1-q) $   \\ 
			$ \beta^4=(0,q,0,1-q) $ 
			
		\end{tabular} & \begin{tabular}{ l }
			
			$ b^m_1(H^+)=1/m, b^m_1(H^-)=1/m^2 $  \\ 
			$ b^m_2(ac)=1/m, b^m_2(buy^+)=1-1/m$   \\ 
			$ b^m_2(buy^-)=b^m_2(buy^0)=1/m $   \\ 
			
		\end{tabular}  \\\hline
		VIII & \begin{tabular}{ l }
			\\
			$ \beta^1=(q,0,1-q,0) $  \\ 
			$ \beta^2=(q,1-q)$   \\ 
			$ \beta^3=(q,1-q) $   \\ 
			$ \beta^4=(q,0,1-q,0) $ 
			
		\end{tabular} & \begin{tabular}{ l }
			
			$ b^m_1(H^+)=b^m_1(H^-)=1-1/m $  \\ 
			$ b^m_2(ac)=b^m_2(buy^+)=1/m$   \\ 
			$ b^m_2(buy^-)=b^m_2(buy^0)=1/m $   \\ 
			
		\end{tabular}  \\\hline
		X & \begin{tabular}{ l }
			\\
			$ \beta^1=(q,0,1-q,0) $  \\ 
			$ \beta^2=(q,1-q)$   \\ 
			$ \beta^3=(1,0) $   \\ 
			$ \beta^4=(q,0,1-q,0) $ 
			
		\end{tabular} & \begin{tabular}{ l }
			
			$ b^m_1(H^+)=1-1/m, b^m_1(H^-)=1-1/m^2 $  \\ 
			$ b^m_2(ac)=1/m, b^m_2(buy^-)=1-1/m$   \\ 
			$ b^m_2(buy^+)=b^m_2(buy^0)=1/m $   \\ 
			
		\end{tabular}  \\
		\hline
	\end{tabular}
	\caption{Belief systems for equilibria in Model~1 under ${\bf A1' }$, and ${\bf A2' }$: For the sequence of
		assessments, the corresponding Bayesian consistent beliefs are
		calculated as in (A.1)-(A.4).}
	\label{table:assessmod1B}
\end{table}

Reasoning for the non-sequential equilibria in Model~1:

\begin{itemize}
	\item \textbf{Equilibrium I} For any $ b^m_1(H^+), b^m_1(H^-) \in (0, 1)$, $ \mathbb{E}_2[\mbox{buy}^0] < \mathbb{E}_2[\mbox{nb}^0]$ as 
	$ \mathbb{E}_2[\mbox{buy}^0]=(u-p)\beta^4_1+(u-p)\beta^4_2+ (-p)\beta^4_3+(-p)\beta^4_4=(u-p)q-p(1-q)=uq-p < 0=\mathbb{E}_2[\mbox{nb}^0].$
	By sequential rationality, this equilibrium cannot be a sequential equilibrium.
	\item \textbf{Equilibrium III and IV} For any completely mixed strategies $ b^m $ with $s_m=b^m_1 (H^+) \rightarrow 0, t_m=b^m_1 (H^-)\rightarrow 0$, consider the corresponding Bayesian consistent belief at the third information set as in (A.3): $ \beta^{m3} \longrightarrow (q,1-q)$.
	By sequential rationality, Player 2 plays $ nb^-$ at his third information set as $ \mathbb{E}_2[\mbox{buy}^-]=(u-p-x)q+(-p-x)(1-q)= uq-p-x <-x=\mathbb{E}_2[\mbox{nb}^-] $.   
	\item \textbf{Equilibrium VII and IX} For any completely mixed strategies $ b^m $ with $s_m=b^m_1 (H^+) \rightarrow 1, t_m=b^m_1 (H^-)\rightarrow 1$, consider the corresponding Bayesian consistent belief at the second information set as in (A.2):
	$ 	\beta^{m2} \longrightarrow (q, 1-q) $.
	By sequential rationality, Player 2 plays $ nb^+ $ at his second information set as $ \mathbb{E}_2[\mbox{buy}^+]=(u-p-x)q+(-p-x)(1-q)= uq-p-x <-x=\mathbb{E}_2[\mbox{nb}^+] $.
\end{itemize}







\begin{table}
	\centering
	\begin{tabular}{ |p{1cm}||p{3cm}|p{3cm}|p{1cm}|  }
		\hline
		\multicolumn{4}{|c|}{NE in Model 2 under $ A1' $, $ A2' $, and $ A3 $} \\ 
		\hline
		& Player 1 & Player 2 & SE\\
		\hline
		I   & $reg^+dis^-$    &$ac~buy^+nb^-buy^0$&  $ \times $ \\
		II&  $reg^+dis^-$ & $ac~buy^+nb^-nb^0$   & $ \checkmark $\\
		III &$dis^+dis^-$ & $ref~buy^+buy^-nb^0$&  $ \times $\\
		IV    &$dis^+dis^-$ & $ref~nb^+buy^-nb^0$&  $ \times $\\
		V&   $dis^+dis^-$  & $ref~nb^+nb^-nb^0$&$ \checkmark $\\
		VI& $dis^+dis^-$  & $ref~buy^+nb^-nb^0$   &$ \checkmark $\\
		\hline
	\end{tabular} 
	\caption{Sequential equilibria in Model~2 under ${\bf A1' }$, ${\bf A2' }$, and ${\bf A3 }$}
	\label{Seq.eq.:Model2B}
\end{table}

\begin{table}
	\centering
	\begin{tabular}{ | c || l | p{7cm} |}
		\hline
		Equilibrium & Belief System & Assessment Sequence ($ m\geq2 $) \\ \hline
		II & \begin{tabular}{ l }
			\\
			$ \beta^1=(q,0,0,1-q) $  \\ 
			$ \beta^2=(1,0)$   \\ 
			$ \beta^3=(0,1) $   \\ 
			$ \beta^4=(q,0,0,1-q) $ 
			
		\end{tabular} & \begin{tabular}{ l }
			
			$ b^m_1(reg^+)=1-1/m, b^m_1(reg^-)=1/m $  \\ 
			$ b^m_2(ac)=b^m_2(buy^+)=1-1/m$   \\ 
			$ b^m_2(buy^-)=b^m_2(buy^0)=1/m $   \\ 
			
		\end{tabular}  \\ \hline
		V & \begin{tabular}{ l }
			\\
			$ \beta^1=(0,q,0,1-q) $  \\ 
			$ \beta^2=(1,0)$   \\ 
			$ \beta^3=(q,1-q) $   \\ 
			$ \beta^4=(0,q,0,1-q) $ 
			
		\end{tabular} & \begin{tabular}{ l }
			
			$b^m_1(reg^+)=1/m, b^m_1(reg^-)=1/m^2 $  \\ 
			$ b^m_2(ac)=b^m_2(buy^+)=1/m$   \\ 
			$ b^m_2(buy^-)=b^m_2(buy^0)=1/m $   \\ 
			
		\end{tabular}  \\ \hline
		VI & \begin{tabular}{ l }
			\\
			$ \beta^1=(0,q,0,1-q) $  \\ 
			$ \beta^2=(1,0)$   \\ 
			$ \beta^3=(q,1-q) $   \\ 
			$ \beta^4=(0,q,0,1-q) $ 
			
		\end{tabular} & \begin{tabular}{ l }
			
			$ b^m_1(reg^+)=1/m, b^m_1(reg^-)=1/m^2 $  \\ 
			$ b^m_2(ac)=b^m_2(buy^-)=1/m$   \\ 
			$ b^m_2(buy^+)=1-1/m, b^m_2(buy^0)=1/m $   \\ 
			
		\end{tabular}  \\\hline
		
		\hline
	\end{tabular}
	\caption{Belief systems for equilibria in Model~2 under ${\bf A1' }$, ${\bf A2' }$, and ${\bf A3 }$: For the sequence of
		assessments, the corresponding Bayesian consistent beliefs are
		calculated as in (A.1)-(A.4).}
	\label{table:assessmod2B}
\end{table}



Non-sequential equilibria in Model~2:

\begin{itemize}
	\item \textbf{Equilibrium I:} For any $ b^m_1(reg^+), b^m_1(reg^-) \in (0, 1)$, $ \mathbb{E}_2[\mbox{buy}^0] < \mathbb{E}_2[\mbox{nb}^0]$ as 
	$ \mathbb{E}_2[\mbox{buy}^0]=(u-p)\beta^4_1+(u-p)\beta^4_2+ (-p)\beta^4_3+(-p)\beta^4_4=(u-p)q-p(1-q)=uq-p < 0=\mathbb{E}_2[\mbox{nb}^0].$
	By sequential rationality, this equilibrium cannot be a sequential equilibrium.
	\item \textbf{Equilibrium III and IV:} For any completely mixed strategies $ b^m $ with $s_m=b^m_1 (reg^+) \rightarrow 0, t_m=b^m_1(reg^-)\rightarrow 0$, consider the corresponding Bayesian consistent belief at the third information set as in (A.3): $ \beta^{m3} \longrightarrow (q,1-q)$.
	By sequential rationality, Player 2 plays $ nb^-$ at his third information set as $ \mathbb{E}_2[\mbox{buy}^-]=(u-p-x)q+(-p-x)(1-q)= uq-p-x <-x=\mathbb{E}_2[\mbox{nb}^-] $.   
	
\end{itemize}









\section{Determination of sequential equilibria when ${\bf uq-p=0}$}
\label{sec:zero.exp.ut.}

Under the assumption $uq-p=0 $, i.e., the uninformed buyer is indifferent
between buying and not buying, Model~1 has 18 pure strategy Nash equilibria
that can be found via the normal-form representation of that simultaneous move
game in Table~\ref{matrix:Model1}. Similarly, there are 10 pure strategy Nash
equilibria of Model~2 via Table~\ref{matrix:Model2}. What is different to the
previous sections is that all Nash equilibria are also sequential
equilibria. Moreover, the set of sequential equilibria in either model is the
union of the sets of sequential equilibria given in \ref{sec:seq} and~\ref{sec:eq.comp:Model12}. 

We present for each model separately the sets of equilibria in Tables~\ref{Seq.eq.:Model1C}, \ref{Seq.eq.:Model2C}, and demonstrate
that they are sequential.


% \subsection{Model 1.3}
% \label{eq.comp:Model1.3}


\begin{table}[h]
	\centering
	\begin{tabular}{ |p{1cm}||p{3cm}|p{3cm}|p{1cm}|  }
		\hline
		\multicolumn{4}{|c|}{NE in Model~1 when $ uq=p $} \\ 
		\hline
		& Player 1 & Player 2 & SE\\
		\hline
		I   & $H^+L^-$    &$ac~buy^+nb^-buy^0$&  $ \checkmark $ \\
		&   & $ac~buy^+nb^-nb^0$   & $ \checkmark $\\ \hline
		II &$L^+ L^-$ & $ref~buy^+buy^-buy^0$&  $ \checkmark $\\
		& & $ref~nb^+buy^-buy^0$&  $ \checkmark $\\
		&    & $ref~nb^+nb^-buy^0$&$ \checkmark $\\
		&  & $ref~buy^+nb^-buy^0$   &$ \checkmark $\\
		& & $ref~buy^+buy^-nb^0$&  $ \checkmark $\\
		& & $ref~nb^+buy^-nb^0$&  $ \checkmark $\\
		&  & $ref~nb^+nb^-nb^0$&$ \checkmark $\\
		&   & $ref~buy^+nb^-nb^0$   &$ \checkmark $\\ \hline
		III& $H^+H^-$  & $ref~buy^+buy^-buy^0$&$ \checkmark $\\
		&  & $ref~nb^+nb^-buy^0$&$ \checkmark $\\
		& & $ref~buy^+nb^-buy^0$&$ \checkmark $\\
		&   & $ref~nb^+buy^-buy^0$&$ \checkmark $\\
		& & $ref~buy^+buy^-nb^0$&$ \checkmark $\\
		&   & $ref~nb^+nb^-nb^0$&$ \checkmark $\\
		&   & $ref~buy^+nb^-nb^0$&$ \checkmark $\\
		&   & $ref~nb^+buy^-nb^0$&$ \checkmark $\\
		\hline
	\end{tabular} 
	\caption{The sequential equilibria of Model~1 when ${\bf uq=p }$}
	\label{Seq.eq.:Model1C}
\end{table}



Before going into the details of why all equilibria are sequential, an
observation regarding to the equilibrium behavior of Player 2 is worth noting:
Unlike  \ref{sec:seq} and~\ref{sec:eq.comp:Model12}%sec:neg.exp.uti.}
, Player 2 is
indifferent between the alternatives at the fourth information set, i.e.,
$ \mathbb{E}_2[\mbox{buy}^0] = \mathbb{E}_2[\mbox{nb}^0]$ for any $ b,\beta $
as
$ \mathbb{E}_2[\mbox{buy}^0]=(u-p)\beta^4_1+(u-p)\beta^4_2+
(-p)\beta^4_3+(-p)\beta^4_4=(u-p)q-p(1-q)=uq-p= 0=\mathbb{E}_2[\mbox{nb}^0].$

\begin{itemize}
\item \textbf{Equilibrium I:} Consider the following completely mixed behavioral strategies:  $b^m_1 (H^+) =1-1/m \rightarrow 1, b^m_1 (H^-)=1/m \rightarrow 0 $, and $ b^m_2 (ac)=1-1/m \rightarrow 1,~b^m_2 (buy^+) =  1 - 1/m \rightarrow 1, b^m_2 (buy^-)= \frac{1}{m} \rightarrow 0, b^m_2 (buy^0) \in (0,1)$. The corresponding Bayesian consistent belief system is constructed as in \ref{sec:seq} as follows:	$
\beta^{m1} \rightarrow (q, 0, 0, 1-q):=\beta^1,
\beta^{m2} \rightarrow (1, 0):=\beta^2,	\beta^{m3} \rightarrow (0, 1):=\beta^3, \beta^{m4} \rightarrow (q, 0, 0, 1-q):=\beta^4.$ 

\item \textbf{Equilibrium II:}  Consider the following completely mixed behavioral strategies:  $b^m_1 (H^+) = b^m_1 (H^-)=\frac{1}{m} \rightarrow 0 $, and $ b^m_2 (ac)=\frac{1}{m}\rightarrow 0,~b^m_2 (buy^+) , b^m_2 (buy^-) , b^m_2 (buy^0) \in (0,1)$ for any $ m\geq 2 $. The associated beliefs with $b^m$ satisfying Bayesian consistency are $ \beta^{m1} \rightarrow (0, q, 0, 1-q):=\beta^1, \beta^{m2} \rightarrow (q, 1-q):=\beta^2, \beta^{m3} \rightarrow (q, 1-q):=\beta^3 \text{and} \beta^{m4} \rightarrow (0, q, 0, 1-q):=\beta^4$. Then, Player 2 is indifferent at all information sets except the first one. 
\item \textbf{Equilibrium III:} Consider the following completely mixed behavioral strategies:  $b^m_1 (H^+) = b^m_1 (H^-)=1 - \frac{1}{m} \rightarrow 1 $, and $ b^m_2 (ac)=\frac{1}{m}\rightarrow 0,~b^m_2 (buy^+), b^m_2 (buy^-) , b^m_2 (buy^0)\in (0,1)$ for all $ m\geq 2 $. The associated beliefs with $b^m$ satisfying Bayesian consistency are $ \beta^{m1} \rightarrow (q, 0, 1-q, 0):=\beta^1, \beta^{m2} \rightarrow (q, 1-q):=\beta^2, \beta^{m3} \rightarrow (q, 1-q):=\beta^3 \text{and} \beta^{m4} \rightarrow (q, 0, 1-q, 0):=\beta^4$. Again, Player 2 is indifferent at all information sets except the first one. 
\end{itemize}



\begin{table}[h]
\centering
\begin{tabular}{ |p{1cm}||p{3cm}|p{3cm}|p{1cm}|  }
	\hline
	\multicolumn{4}{|c|}{NE in Model 2 when $ uq=p $} \\ 
	\hline
	& Player 1 & Player 2 & SE\\
	\hline
	I   & $reg^+dis^-$    &$ac~buy^+nb^-buy^0$&  $ \checkmark $ \\
	&  $reg^+dis^-$ & $ac~buy^+nb^-nb^0$   & $ \checkmark $\\ \hline
	II &$dis^+dis^-$ & $ref~buy^+buy^-buy^0$&  $ \checkmark $\\
	&$dis^+dis^-$ & $ref~nb^+buy^-buy^0$&  $ \checkmark $\\
	&   $dis^+dis^-$  & $ref~nb^+nb^-buy^0$&$ \checkmark $\\
	& $dis^+dis^-$  & $ref~buy^+nb^-buy^0$   &$ \checkmark $\\
	&$dis^+dis^-$ & $ref~buy^+buy^-nb^0$&  $ \checkmark $\\
	&$dis^+dis^-$ & $ref~nb^+buy^-nb^0$&  $ \checkmark $\\
	&   $dis^+dis^-$  & $ref~nb^+nb^-nb^0$&$ \checkmark $\\
	& $dis^+dis^-$  & $ref~buy^+nb^-nb^0$   &$ \checkmark $\\
	\hline
\end{tabular} 
\caption{The sequential equilibria of Model~2 when ${\bf uq=p }$}
\label{Seq.eq.:Model2C}
\end{table}


The observation which was done above for Model 1 remains valid for Model 2, i.e., Player 2 is indifferent between the alternatives at the fourth information set.
\begin{itemize}
\item \textbf{Equilibrium I:}
Consider $b^m_1 (reg^+) = 1 - \frac{1}{m} \rightarrow 1 $, $b^m_1 (reg^-) = \frac{1}{m} \rightarrow 0$, $b^m_2 (ac) = 1 - \frac{1}{m} \rightarrow 1$, $b^m_2 (buy^+) = 1 - \frac{1}{m} \rightarrow 1$, $b^m_2 (buy^-) = \frac{1}{m} \rightarrow 0$ and 
$b^m_2 (buy^0) \in (0,1) $. Based on this behavioral strategies, Bayesian consistency implies $ \beta^m $, the computed beliefs above: $ \beta^{m1} \rightarrow (q, 0, 0, 1-q):=\beta^1,
\beta^{m2} \rightarrow (1, 0):=\beta^2,
\beta^{m3} \rightarrow (0, 1):=\beta^3,
\beta^{m4} \rightarrow (q, 0, 0, 1-q):=\beta^4. $ 

\item \textbf{Equilibrium II:}
Consider $ 	b^m_1 (reg^+) = b^m_1 (reg^-) = \frac{1}{m} \rightarrow 1,
b^m_2 (ac) = \frac{1}{m} \rightarrow 0,
b^m_2 (buy^+),\\
b^m_2 (buy^-),
b^m_2 (buy^0) \in (0,1) $. These strategies lead to the Bayesian consistent beliefs in the way described above, so $ \beta^{m1} \rightarrow (0, q, 0, 1-q):= \beta^1 ,
\beta^{m2} \rightarrow (q, 1-q):= \beta^2,
\beta^{m3} \rightarrow (q, 1-q):= \beta^3, \beta^{m4} \rightarrow (0, q, 0, 1-q):= \beta^4. $ 

\end{itemize}










\end{appendices}



\bibliography{sn-bibliography}
\end{document}
