10

In a Bayesian game, information is incomplete. To cope, players have beliefs about the state of the game. In a sense, each player strategizes as if the game was as he or she believes. So each player operates in his or her own world. And if every player plays a Nash equilibrium in one's own world, that's a Bayesian Nash equilibrium. In a stochastic game, the ...


9

I believe that the answer given by @denesp is incorrect. The second method involves simply writing the game in strategic of "normal" form. I believe the first method is better (easier to use), but I think that they can both be used. In the answer given by @desesp, the following explanation is given. The reason why method two is flawed is that the ...


7

Epistemic game theory would be the closest (sub-)field that deals with questions involving higher order beliefs among interacting agents. The introductory article by Dekel and Siniscalchi is a good entry point to the literature. From its introduction: Epistemic game theory formalizes assumptions about rationality and mutual beliefs in a formal language, ...


6

It is actually assumed that $b_i(v_i)$ is of the form $\alpha_i+\beta_i \cdot v_i$. So it is an affine function. Linearity only works if the bottom of the uniform distribution is 0. A somewhat intuitive reasoning is that if the valuations are uniformly distributed over $[c,d]$ and $b_i(v_i)$ is a symmetric equilibrium, then $b_i(v_i)+k$ should define a ...


6

After posting a bad solution yesterday I believe I got a better one: The strategy of the buyer consists of two functions, $(f_1(v,p_1),f_2(v,p_1,p_2))$ where both functions map to $\left\{A,R\right\}$ (where $A$ stands for Accept, $R$ for Reject). The strategy of the seller is $(p_1,p_2(f_1(v,p_1)))$. You get the solution via backward induction. In PBE $f_2(...


6

Signaling is the informed side taking actions to reduce (or maintain, depending on the private types) the information asymmetry. For example, high skill workers getting certifications to signal their productivity so as earn a higher wage. Or low skill workers trying to mimic the high-skilled's behavior as much as possible so that they cannot be separated. ...


5

Bayesian Nash equilibrium is a set of strategies $\{\sigma_i\}$ one for each player and some beliefs $\{\mu_i\}$ also one for each player such that $\sigma_i$ is a best response for player $i$ given his belief, $\mu_i$, and the beliefs are Bayesian for all players, given their information. Each strategy $\sigma_i$ is a function from the set of types to a (...


5

While it is a bit unusual to describe a strategy profile as being Pareto optimal, especially in the context of Bayesian games, I guess you can still define Pareto optimality in different stages of such games as follows. Recall that in a Bayesian game, a pure strategy is a function $s_i:\Theta_i\to A_i$. A strategy profile $s(\theta)=(s_1(\theta_1),\dots,...


5

There is no "universal Aumann model", as shown in Heifetz & Samet, GEB 1998, "Knowledge Spaces with Arbitrarily High Rank ", even though there is a universal type space. On a less technical level, Aumanns model does not allow for wrong beliefs. A generalization, so called Kripke structures, do however.


5

I would suggest you start by looking at C. Dellarocas. "The Digitization of Word-of-Mouth: Promise and Challenges of Online Reputation Systems". Management Science 49 (10), October 2003, 1407-1424. for a review of relevant literature and Friedman & Resnick "The Social Cost of Cheap Pseudonyms". Journal of Economcs and Management Strategy. 10 ...


5

In a standard cheap-talk setting, a sender (S) has better information on a state of the world and wants to communicate this information to a receiver (R) who then takes an action. However, S and R prefer different actions conditional on the state. Importantly, S is free to send any message independent of what she knows. That is, she cannot commit to a signal ...


5

You could represent the game in extensive form like this: The dashed lines enclose player 2's information set. This encompasses all of player 2's nodes because player 2 observes neither nature's nor player 1's choice. Player 1's information sets are the two singleton nodes because player 1 knows which branch nature has chosen.


5

$U^S(\bar y,\bar m,b)=\max_{y\in Y}U^S(y,\bar m,b)$ is standard notation that says $\bar y$ is the action (taken by receiver) that would maximize sender's utility given message $\bar m$ and bias parameter $b$. In other words, sender would prefer receiver to choose $\bar y$ when he sends message $\bar m$. Sender's maximized utility is $U^S(\bar y,\bar m,b)$. ...


4

I think your definition is incorrect, or at least incomplete.** Usually, in a Bayesian game, there is assumed to be a prior distribution on $T$ (where $T = \times_i T_i$). This distribution is called the "common prior" and it is assumed to be common knowledge that types are drawn according to this distribution. In this case, each player $i$'s belief $p_i$ is ...


4

Yes, a whole book has been written on Behavioral Game Theory. More specifically, standard solution concept such as Nash equilibrium requires that players best respond to a correct belief about other players' moves. The following are examples that relax one of these cognitive restrictions: Quantal response equilibrium allows for the possibility that ...


4

Sure. In the two type case there is an alternate solution that is frequently used. If $a \neq c$ and $b \neq d$, that is the types are really different in both attributes, then you can define a function $f$ for which $$ b = f(a), \hskip 20pt d = f(c). $$ Whatever you need the second attribute for, this way it becomes a function of the first one, and the ...


4

Yes, you are correct. All types $t_{1}$ choose O (B) and all types $t_{2}$ choose O (B) are both Bayesian equilibria. Note that there are other Bayesian equilibrium in this game, if you are interested this is explained in detail here (p. 10, see reference below) for this particular battle of the sexes with two-sided incomplete information. The basic idea is ...


4

To understand the "Obedience" inequality, notice that the player is "integrating out" everything that she is uncertain about, this is why the sum runs over the actions of other players, $a_{-i}$, the types of other players, $t_{-i}$, and the state of the world, $\theta$. The first term simply gives the probability that the mediator observed $\theta$. For ...


4

The Key to BNE is that players that know something (about the state of the world or their type) can condition their strategies with their information. That is, for example, in question 3, type A could choose strategy U, while type B could choose strategy D. Therefore from the second player's perspective, there are four possible pure strategies of his ...


4

The solution concept of Bayes Correlated Equilibrium applies to games, viz strategic interactions, between multiple players. Thus, in a single person decision problem its use seems to me inappropriate or at least superfluous. A lot of effort has been spent over the last 70 years (dating back at least to Blackwell 1951, 1953) to explore the notion of ...


4

Claim: If choice sets $T, M,$ and $A$ are finite, then an assessment $\{\beta^*_{r}, \beta^*_{s}, \mu^*\}$ is a WPBE (weak perfect Bayesian equilibrium) of the two-stage signalling game between receiver $r$ and sender $s$ if and only if it is a SE (sequential equilibrium). Proof: SE $\implies$ WPBE is trivial since SEs are PBEs by construction, and thus are ...


3

If $10^{10}$ pure strategies is too large for Gambit, it'll likely be too large for any other software as well. In the comments, I suggested that you could first compute the expected payoffs for each player in a separate program such as Matlab/R/Excel, and then export the resulting matrix to Gambit where the BNEs can be calculated. This way, you may be able ...


3

Sure. An example: if both valuations are drawn from the $[0,1]$ interval then the strategies $$ b_1(v_1) = v_1 $$ and $$ b_2(v_2) = \left\{\begin{array}{cc} v_2 & \text{ if } v_2 < 1 \\ 5 & \text{ if } v_2 = 1. \end{array}\right. $$ Another, slightly more annoying equilibrium for $v_1,v_2 \in [0,1]$: $$ \begin{align*} b_1(v_1) & = 0 \\ b_2(v_2)...


3

It's basically what Keynes called the «beauty contest». Back then the newspaper offered rewards to the person that was able to guess who where the pretiest girls (determined by the poll). My english is bad (french student). hope i was a little help.


3

In some ways, beliefs are a more natural way of interpreting solution concepts like Nash equilibrium and its refinements. For example, consider a simple simultaneous-move coordination game as follows: \begin{equation} \begin{array}{c|cc} &L&R\\\hline T&1,1&0,0\\ D&0,0&2,2 \end{array} \end{equation} We say that $(D,R)$ is a Nash ...


3

Signaling games (games in which an informed "sender" moves first and an uninformed "receiver" second) typically have a plethora of Perfect Bayesian Equilibria which is not really appealing in terms of predictive power. However, as you already said, some equilibria may be "unreasonable". Refinements serve the purpose to formalize which of these equilibria are ...


3

Let $BR=(BR_1,BR_2,BR_3,...)$ denote the best response mapping. This gives a mapping given message vector $m$. (For any $BR_i$ only the messages of non $i$ players are used to derive the best response mapping.) Let $m^* = (m_1^*,m_2^*,m_3^*,...)$ denote the symmetric equilibrium. As it is an equilibrium, for any $i$ you have $BR_i(m^*) = m_i^*$, so $$ BR(m^*...


3

I think the reason you're having difficulty is that your definition is actually not equivalent to Bergemann and Morris' definition of BCE, except under specific assumptions which include the ones you state regarding independence. However, in general, we do not wish to make these assumptions. (For example, assuming that $t_{-i}$ is independent of $a_i$ ...


3

Its for all priors and all utility matrices. Its possible for $p_1,u_1$ and $p_2,u_2$ to rank $S$ and $S'$ different.


3

In the second period, the buyer accepts any offer $s_{2}$ $\leq$ $0.7$ if he has rejected the first offer and any offer $s_{2}$ $\leq$ $0.3$ if he has accepted the first offer. Given this, there are only two offers that may be optimal for the second seller: either $s_{2}$ $=$ $0.3$ or $s_{2}$ $=$ $0.7$. Let $\mu$ denote the probability that the second seller ...


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