<p>This is a static Bayesian game. The two players share a common prior over the type space which is characterized by the uniform distribution. For Bayesian games, you have to specify the belief of each player given his/her type, and the belief of each player determines their action. Basically, the strategy is a mapping from one's type to an action through belief specification.  </p>

<p>What is player <span class="math-container">$i$</span>'s belief about <span class="math-container">$v_j$</span>? It is independently drawn from the uniform distribution over <span class="math-container">$[0,1]$</span>, so <span class="math-container">$E(v_j) = 0.5$</span> for now. And he will exchange only when <span class="math-container">$v_i \leqslant \frac{3}{2}v_j$</span>, but he does not know exactly the value of <span class="math-container">$v_j$</span>, so he takes an expectation over <span class="math-container">$v_j$</span>, then as long as <span class="math-container">$v_i \leqslant \frac{3}{2} \times 0.5 = \frac{3}{4}$</span>, he will choose to exchange. </p>

<p>The same goes for player <span class="math-container">$j$</span> because they are symmetric, so player <span class="math-container">$j$</span> will only exchange when <span class="math-container">$v_j \leqslant \frac{3}{4}$</span>.  </p>

<p>Given this, <span class="math-container">$E(v_j)$</span> now decreases to <span class="math-container">$\frac{3}{8}$</span>, player <span class="math-container">$i$</span>'s expected payoff from exchange decreases to <span class="math-container">$\frac{3}{2}\times \frac{3}{8} = (\frac{3}{4})^2$</span>. Therefore, player <span class="math-container">$i$</span> will only choose exchange when <span class="math-container">$v_i \leqslant (\frac{3}{4})^2$</span>.</p>

<p>Since <span class="math-container">$\frac{3}{4} &lt; 1$</span>, this process will go on and on such that no one will choose exchange in the end which is the only equilibrium. </p>

CC BY-SA 4.0
2020-05-27T14:30:13.593Z
2020-05-27T15:02:14.773Z
2020-05-27T15:02:14.773Z
Lin Jing
<p>My question concerns the following problem: two players, <span class="math-container">$1$</span> and <span class="math-container">$2$</span>, each owns a house. Each player <span class="math-container">$i$</span> values his own house at <span class="math-container">$v_{i}$</span>. The value of player <span class="math-container">$i$</span>'s house to the other player, i.e. to player <span class="math-container">$j\neq i$</span>, is <span class="math-container">$\frac{3}{2}v_{i}$</span>. Each player <span class="math-container">$i$</span> knows the value <span class="math-container">$v_{i}$</span> of his own house to himself, but not the value of the other player's house. The values <span class="math-container">$v_{i}$</span> are drawn independently from the interval <span class="math-container">$[0,1]$</span> with uniform distribution. </p>

<p>Payoffs and actions are defined as follows: each player announces simultaneously whether they want to exchange their houses. If both players agree to an exchange, the exchange takes place. Otherwise no exchange takes place. </p>

<p>As far as finding a Bayesian equilibrium is concerned, I have reached the following stage:</p>

<p>Given <span class="math-container">$j$</span> exchanges, <span class="math-container">$i$</span> exchanges so long as <span class="math-container">$v_{i}\leq\frac{3}{2}v_{j}$</span> and given <span class="math-container">$i$</span> exchanges, <span class="math-container">$j$</span> exchanges so long as <span class="math-container">$v_{j}\leq\frac{3}{2}v_{i}$</span>. This means that <span class="math-container">$j$</span>'s expected utility from exchanging (given <span class="math-container">$i$</span> exchanges) is <span class="math-container">$\frac{9}{8}v_{j}$</span> and <span class="math-container">$i$</span>'s is <span class="math-container">$\frac{9}{8}v_{i}$</span>. From this I do not know how to proceed. </p>

CC BY-SA 4.0
2020-05-27T07:34:28.170Z
@Giskard Thanks. I have edited the question.
Charles
CC BY-SA 4.0
2020-05-27T11:52:15.203Z
Why do you think that j's expected utility from exchanging (given i exchanges) is $\frac{9}{8}v_j$ ?
VARulle
<microeconomics><game-theory><mathematical-economics>
Tadelis ‘trading places’ Bayesian game
Charles
# Tadelis ‘trading places’ Bayesian game

(Asked by: username_0 on 2020-05-27)

My question concerns the following problem: two players, $1$ and $2$, each owns a house. Each player $i$ values his own house at $v\_{i}$. The value of player $i$'s house to the other player, i.e. to player $j\neq i$, is $\frac{3}{2}v\_{i}$. Each player $i$ knows the value $v\_{i}$ of his own house to himself, but not the value of the other player's house. The values $v\_{i}$ are drawn independently from the interval $[0,1]$ with uniform distribution.

Payoffs and actions are defined as follows: each player announces simultaneously whether they want to exchange their houses. If both players agree to an exchange, the exchange takes place. Otherwise no exchange takes place.

As far as finding a Bayesian equilibrium is concerned, I have reached the following stage:

Given $j$ exchanges, $i$ exchanges so long as $v\_{i}\leq\frac{3}{2}v\_{j}$ and given $i$ exchanges, $j$ exchanges so long as $v\_{j}\leq\frac{3}{2}v\_{i}$. This means that $j$'s expected utility from exchanging (given $i$ exchanges) is $\frac{9}{8}v\_{j}$ and $i$'s is $\frac{9}{8}v\_{i}$. From this I do not know how to proceed.

**Tags:** microeconomics, game-theory, mathematical-economics
> **Comments:**
> **username_0**: @Giskard Thanks. I have edited the question.  
> **username_1**: Why do you think that j's expected utility from exchanging (given i exchanges) is $\frac{9}{8}v_j$ ?  

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## 1 answers

### Answer by username_2 on 2020-05-27. Score: 3

This is a static Bayesian game. The two players share a common prior over the type space which is characterized by the uniform distribution. For Bayesian games, you have to specify the belief of each player given his/her type, and the belief of each player determines their action. Basically, the strategy is a mapping from one's type to an action through belief specification.

What is player $i$'s belief about $v\_j$? It is independently drawn from the uniform distribution over $[0,1]$, so $E(v\_j) = 0.5$ for now. And he will exchange only when $v\_i \leqslant \frac{3}{2}v\_j$, but he does not know exactly the value of $v\_j$, so he takes an expectation over $v\_j$, then as long as $v\_i \leqslant \frac{3}{2} \times 0.5 = \frac{3}{4}$, he will choose to exchange.

The same goes for player $j$ because they are symmetric, so player $j$ will only exchange when $v\_j \leqslant \frac{3}{4}$.

Given this, $E(v\_j)$ now decreases to $\frac{3}{8}$, player $i$'s expected payoff from exchange decreases to $\frac{3}{2}\times \frac{3}{8} = (\frac{3}{4})^2$. Therefore, player $i$ will only choose exchange when $v\_i \leqslant (\frac{3}{4})^2$.

Since $\frac{3}{4} < 1$, this process will go on and on such that no one will choose exchange in the end which is the only equilibrium.


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