Questions tagged [bayesian-game]

For questions about Bayesian games. These are strategic interactions when one or more players have incomplete information about other players. General topics of games with asymmetric information may use this tag.

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Coordination BNE

I don't have a lot of experience on BNE so this question got me stumped. Two player simultaneously have to decide whether to go to a movie, i.e. to Show up (S) or not (N). They prefer to go together, ...
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2 votes
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Is the mixed strategy $\sigma_i^*:S_i\times\Theta\to\Delta(A_i)$ chosen by player $i$ linear?

Suppose that we have a Bayesian game, where the number of players is $I$ and we refer to the generic player with $i$ $\Theta$ stands for the state of the world, where $\theta\in\Theta$ is the typical ...
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I present a communication game - Could you please make comments on my assumptions, notation and properties that I may have not considered yet?

I consider the following communication game. Suppose that we have $I$ players and each one of them learns a private signal $s_i=(s_{i,1},s_{i,2},...,s_{i,k})$, where $k$ is finite and also, every ...
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In games of Bayesian Persuasion, under what conditions is Receiver better off than under uninformative signals?

I've been working through a few Bayesian Persuasion (BP) models à la Kamenica and Gentzkow (2011), and a feature that seems to arise often is that commitment to a persuasion mechanism is Pareto ...
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Bayes Nash equilibrium distribution

I would like to define a decision rule that is induced by a BNE distribution in a game with a continuum of agents. For that, I have a decision rule $\varphi:\Theta \rightarrow\Delta(\Delta(A))$ that ...
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1 answer
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How does this reporting correspondence is redefined?

Let $\mathcal{R}_i$ be a non-empty, finite set and define the reporting correspondence $R_i:S→2^{\mathcal{R}_i}-\{\emptyset\}$ to be a mapping from player i’s type space to the collection of subsets ...
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2 votes
1 answer
63 views

Bayes Correlated equilibrium without information expansion

The questions are with reference to Bergemann and Morris (2013, 2016). I'm trying to give an alternative interpretation to BCE from the analyst/information designer perspective provided in the paper. ...
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4 votes
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Perfect Bayesian Equilibium - Application to game with inconsistent beliefs / no common prior

Does the concept of a Perfect Bayesian Equilibrium apply only to incomplete games with a common prior / consistent belief? In both Bonanno's "Game Theory" and Osborne's "A Course in ...
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Are these two definitions of Bayesian Nash Equilibrium equivalent?

Consider a standard game $\Gamma$ with incomplete information. There are $n$ players indexed by $i=1,...,n$. $S_i\equiv \{s_{i1},...,s_{iJ}\}$ is the set of actions of player $i$. $S\equiv \times_{i=1}...
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2 votes
1 answer
61 views

Is the following claim written with the right way?

I have a simple question though confusio for me. In game theory we usully write thet a strategy is a mapping from the set of types $T$ to the simplex set of actions (refering to mixed mixed strategies)...
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Extension of Harsanyi Transform for Two-sided Incomplete Information Games to Beliefs with Zero Probability

In the textbook I'm reading "Game Theory - Giacomo Bonanno", one requirement to applying the Harsanyi transform to convert a two-sided incomplete information game to an imperfect information ...
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Why does the belief over information sets with probability zero matter in Perfect Bayesian Equilibrium?

I'm struggling to understand why the notion of "belief revision" is an important concept. In particular, why does the belief over information sets with probability zero matter? When ...
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Equivalence from correlated/communication equilibrium to Nash Equilibrium?

Taking into account the seminal papers of Forges and Imre Bárány, they proove a very strong result that gives an exact connection among the communication and the correlation equilibrium solution ...
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3 votes
1 answer
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What is the intuition behind Blackwell's Equivalence Theorem on Information Structures?

Let us suppose that we have a Bayesian game where the information structure is defined to be as $P^X=\{(X,\mathcal{X},P_\theta)\}_{\theta\in\Theta}$ where a signal generated by the information ...
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2 votes
1 answer
117 views

Sequential and Perfect Bayesian Equilibrium: an example?

My question is quite simple. Could someone given an example of how to determine a Sequential Equilibrium given a set of Perfect Bayesian Equilibria? The definition of sequential equilibrium where ...
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2 votes
1 answer
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Is playing against state of nature considered Stochastic game or Bayesian game?

Say if there is a team of gamblers betting on stock exchange. We can model the outcome of stock exchange as state of nature, because it is not deterministic. So the objective of team of gamblers is to ...
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What is the difference between Partially Observable and Incomplete information in Game theory

In reference to this paper that discusses Partially Observable Stochastic Game, it also mentions of Bayesian Game which is characterized by incomplete information. Rather then pointing out the ...
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Correlation device that induces a specific transition probability

Taking a look at this paper of Forges and Vida the authors define a correlation device in page $102$, that is a standard probability space $\left(\Omega,\mathcal{B},\mu\right)$, They assume that the ...
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What if Bergemann and Morris setting used mixed (or bbehavioral) actions instead of pure actions as reccomendations?

Once again, I will refer to the setting of Bergemann and Morris (2016) and write here the payoff formula of player $i$ from the perspective of the information designer. The payoff formula is the ...
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2 votes
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Pre-play, interim-play, intra-play communication games

I have a question that seems easy but I want some clarification. Suppose that we have a communication Bayesian game. The game starts when the players learn their prior infornation and after this they ...
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3 votes
1 answer
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Bergemann and Morris information designer and decision rule concept

Taking a look in the paper of Bergemman and Morris in 2016, they refer to the desicion rule as mapping $$\sigma:\Theta\times T\to\Delta(A)$$ The explanation to understand the notion of it is given as ...
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2 votes
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Repeated Games of Incomplete information set up and questions

The following are from the model of Jérôme Renault (Repeated Games of incomplete infomration). The preliminaries of the model Formally, a repeated game with incomplete information is given by the ...
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How to define a dynamic programming problem in an incomplete information game?

How can I define a problem of dynamic programming, to use the Hamilton-Jacobi-Bellman equation in order to solve the utility maximization problem of the generic agent of a dynamic game with incomplete ...
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How does this definition change when we introduce some kind of information?

In Heller et al, they use the Osborne and Rubinstein formal definition for the extensive form games with public information. However, the model can work with the same definitions in case all players ...
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1 answer
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How to analyze this two-player, imcomplete-information, extensive-form game?

Game tree: Note that the information sets for player 1 are represented by red dotted lines, while the information sets for player 2 are represented by blue shaded tapes. Described in words, the game ...
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3 votes
1 answer
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Is it a weakly dominant strategy to bid your true value in a third-price auction with two identical goods and N uniform distributed bidders?

I am interest in finding the optimal strategy for a single bidder (the BNE), when I have a third-price auction with N bidders and two identical goods. The bidders have iid valuations: U(0,1) I have ...
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1 vote
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What is the difference between a communication mechanism and a protocol of communication?

What is the difference between a communication mechanism and a protocol of communication in mechanism and information design. Although most papers in the literature make this distinction, it is not ...
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1 vote
1 answer
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Expected payoff calculation difficulty in the Bayesian environment of Bergemman and Morris

Suppose that we have two states of the world equally likely to occur, and say $\psi$ is the common prior of the state $\theta\in\Theta=\{G,B\}$. The types of the players are given by the following ...
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1 vote
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Bayesian update in the beliefs about the signals

Suppose that we have tow states of the world $\omega_1$ and $\omega_2$, where $p(\omega_1)=p(\omega_1)=1/2$ and there are three different signals, $s_H,s_M,s_L$ that are equally likely to occur in ...
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Information acquisition problem and desing of information

I am searching some details about the information acquisition problem. I want to understand what this is about and where can I find the beginning of this literature. I think this problem is some kind ...
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Questions about the definition of a protocol

The following definition is from this paper about protocols of communications (definition $2.4$ in the paper). You can check the paper for more details and previous of my questions. $\textit{...
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When should my beliefs be a martingale?

Suppose that I am wondering about whether an event will happen (e.g. whether it will rain tomorrow in a particular location). Suppose that my beliefs about whether the event will happen can be ...
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3 votes
1 answer
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Bayes correlated equilibrium of Bergemann and Morris

The paper of Bergemann and Morris proves a theorem based on some foundations about the information sets and their expansions. I am trying to understand theorem one intuition, more precisely I cite the ...
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1 vote
1 answer
81 views

Mechanisms of communication in game theory

In the spirit of the previous question that I have done, here considering the paper here I am trying to make the matching definition $2.2$ here. I will give two definitions and I would like to clarify ...
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3 votes
1 answer
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Difficulty in understanding the notation related from probability theory with game theory

The question that I have is a little technical and it has to do with the notation and the combination between some mathamatical properties in the probability theory of information economics. Say $\...
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2 votes
1 answer
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Alternative way to calculate the symmetric BNE of the game

My problem. Consider the following auction for a single object. There are $n \geq 2$ bidders. They submit their bids simultaneously. The object is allocated to the player who submits the largest bid. ...
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1 vote
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Job Market Signaling

My Solution is this Separating equilibrium High type worker chooses $e_H = e^* >0$ Low type worker chooses $e_L =0$ Firms believes that if education is $e^* >0$ then the worker is high type and ...
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1 vote
2 answers
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Rule for Number of Strategies in Bayesian Games

Is there a general rule for finding the number of strategies (denote as $S$) for each player in a Bayesian game? I think it's related to the number of types (denote as $T$) and the number of actions (...
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5 votes
1 answer
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Bayesian Nash Equilibrium in a Duopoly Cournot Competition

I am having a hard time to solve a Bayesian Nash equilibrium game in a duopoly cournot competition setting. So, I have two firms with given production quantities, let's say $q_1$ and $q_2$ (i.e., not ...
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2 votes
1 answer
98 views

Find a separating perfect Bayesian equilibrium

Exercise Question 2, Chapter 28, Strategy: An Introduction to Game Theory 3rd Edition by Joel Watson In part (a) of the question, we have to check if any separating perfect Bayesian equilibrium ...
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1 vote
1 answer
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Strong sequential equilibria and the existence of others

I am working on the following game and I have to find all strong sequential equilibria here. I determined that here any belief derived from a fully mixed strategy gives a distribution (1/2, 1/2) over ...
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1 vote
1 answer
182 views

Simultaneous vs Sequential Games [closed]

Is there a way to characterize the distinction between simultaneous vs sequential games? I'm trying to describe a situation where players can only take actions without knowledge of other players' ...
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5 votes
0 answers
61 views

How to find an optimal strategy in an auction?

I have asked this question in mathematics forum as well but since I have not recieved an appropriate answer yet, I ask it here as well. Consider an auction of sculptures by four artists: A, B, C and D....
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5 votes
1 answer
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What does 'continuation' as in continuation games, strategies, plays, etc. exactly mean?

I am in my first course in grad level game theory. While I was reading through Fudenberg and Tirole's Game Theory, I constantly come into contact with the word 'continuation' to describe some games ...
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0 votes
1 answer
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Find a Perfect Bayesian Equilibrium with a game that includes Nature

I want to determine all pure-strategy Perfect Bayesian Equilibria for this task, but I cannot get very far. Side note: First number is payoff for A, second number payoff for player B. For the last ...
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1 vote
1 answer
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Exchange Houses game - Bayesian Game

Can anyone help me understand how to solve this type of asymmetric information Bayesian game? So the game is a different version of the Tadelis 'trading house games'. It involves 2 players that each ...
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1 vote
1 answer
336 views

Mixed Strategy Bayesian Nash Equilibrium

There was an exercise question regarding two players with two types each in a game theory class. The two players were assigned to do a team project together. The utility from doing the team project is ...
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3 votes
1 answer
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Mixed Strategies in Bayes Nash Equilibrium (Bayesian Battle of the Sexes). Shouldn't it depend on $p$?

I have a question about calculating mixed strategies in a Bayes Nash Equilibrium in a simple 2-player bimatrix game. To demonstrate the issue, consider ``Bayesian Battle of the Sexes.'' Suppose P1 ...
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2 votes
1 answer
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I cannot understand Sobel & Crawford 1982 (cheap talk)

I am a master student, currently reading this paper for fun during the break. I got lost during the part $U^S(\bar y, \bar m, b) = \max_{y \in Y} U^S(y,\bar m, b)$ on page 5. I dont understand why ...
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2 votes
1 answer
191 views

Pooling equilibrium in akerlof's lemon market with certification cost

Consider a market for 300 used cars where 1/3 of all cars are good quality cars and the rest are bad quality cars. All these cars are owned by (potential) sellers to begin with and each seller owns ...
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