# Tag Info

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 $f_2(... 6 Signaling is the informed side taking actions to reduce (or maintain, depending on the private types) the information asymmetry. For example, high skill workers getting certifications to signal their productivity so as earn a higher wage. Or low skill workers trying to mimic the high-skilled's behavior as much as possible so that they cannot be separated. ... 6 We have that${\cal I} = ((X^i)_i, \mu)$​ and${\cal J} = ((Y^i)_i, \nu)$​ are two information structures. An Interpretation mapping for player$i$​​ is a mapping$\phi^i: X^i \to \Delta(Y^i)$​ so it associates with every$x^i$​ a distribution over$Y^i$​. Let$x^i \in X^i$​. Then$\phi^i(x^i)$​ is a distribution over$Y^i$​ so$\phi(x^i)(y^i)$​ is the ... 5 While it is a bit unusual to describe a strategy profile as being Pareto optimal, especially in the context of Bayesian games, I guess you can still define Pareto optimality in different stages of such games as follows. Recall that in a Bayesian game, a pure strategy is a function$s_i:\Theta_i\to A_i$. A strategy profile$s(\theta)=(s_1(\theta_1),\dots,...

5

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

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

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 ...

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 $U^S(\bar y,\bar m,b)$. ...

5

The expected posterior is the prior. But for this to hold, you have to take the expectation with respect to the prior. In your example, you describe a Bayesian but ev aluate their update with respect to your belief. Ann and Bob might be perfect Bayesians and their expected posteriors might be exactly their priors. But that does not mean that Ann's ...

5

The common prior $p\in\Delta L$ and the transition probability $q:L\to\Delta A$ induce a joint distribution on $L\times A$ in which the pair $(l,a)$ is selected with probability $p_l\cdot q_l(a)$. You can also recover $q$ outside (strategically irrelevant) type profiles $l$ such that $p_l$ by the usual formula for conditional probabilities. Now, an ...

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

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)... 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 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 ... 4 Consider a game with private information such as a privately known willingness-to-pay or any other type. We usually model this as a game in which at first "Nature" draws the type and then players make their moves. Such games do not have proper subgames because a proper subgame never splits up an information set and Nature's first move connects the ... 4 There are three classes of equilibria of this game. The first class is sequential: $$(s_1,s_2)=(y,r)$$ and the beliefs are $$\mu_1(a)=\mu_1(b)=\mu_2(a\mid y)=\mu_2(b\mid y)=\frac12.$$ The second class is not sequential, but weak perfect Bayesian: $$(s_1,s_2)=(x,l)$$ and the ... 4 If the capacities q_1 and q_2 are the same, say \overline{q} you can rule out some cases. The best response functions are given by: \begin{align*} \hat q_1 &= \min\left\{\frac{M - (1-\theta) \hat q_2^L - \theta \hat q_2^H - m}{2}, \overline{q}\right\},\\ \hat q_2^L &= \min\left\{\frac{M - \hat q_1 - m}{2}, \overline{q}\right\},\\ \hat q_2^H &...

4

There is another way to compute the symmetric BNE in increasing strategy. Let $U(v)$ denote the expected utility of a player in equilibrium when his type is $v$: Given that the bidding strategy is increasing, a player with type $0$ will get the good with probability zero. Thus he/she must bid zero and $U(0) = 0$. For any other $v > 0$, the probability ...

4

Suppose you are an analyst studying a Bayesian game. You know the players, the possible states of nature, the common prior, the action spaces, the payoff functions, and you know about some information channels available to the players, the latter given via some information structure. However, you can't rule out that the players have additional information ...

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