# Arguing uniqueness of Bayes-Nash equilibrium in an auction setting

In an auction setting with interdependent values, let $\theta_i$ denote the type of player $i$ and $m_i$ that player's message (a bid, essentially). I have calculated the best response function as: $$m_i^* = \theta_i + \gamma \sum_{j \ne i} (\theta_j-m_j)$$ Obviously, then, truth-telling by all parties ($m_k = \theta_k$ for all $k$) is a symmetric BNE, regardless of $\gamma$. I want to also argue that it is the unique BNE. If $\gamma = 1$, it seems like there would be assymetric BNE if there are only two players in which one player's positive misreport ($m_i - \theta_i$) would equal the other player's negative misreport ($\theta_j - m_j$). If $\gamma<1$, it looks like the truth-telling BNE is unique, but how would I prove that?

Also, so far I have not used a common prior assumption. Do I need it here to argue the existence of the truth-telling BNE, or to argue it's uniqueness? If so, we can suppose $\theta_i$ is drawn i.i.d. uniformly from $[0,1]$.

• I would like to use this model to demonstrate a technique to someone. Could you give me a reference or let me know where I can find the model from which you arrived at this best response function? Feb 12, 2016 at 10:13
• Have a look at Robust Implementation in Direct Mechanisms, 2009, by Bergemann and Morris. I should mention that your argument below is complete only if the action spaces are unbounded. If they're bounded, you're technically just proving that no interior non-truth-telling BNE exists. In that setting, the only way I know how to get uniqueness is through a contraction property, as you proposed originally. The BR above is a contraction if $\gamma<\frac{1}{n-1}$, with $n$ the number of players. Feb 13, 2016 at 13:53
• Thank you for the paper. Having just read I would like to point out that Bergemann and Morris also look at the cases where $\gamma < \frac{1}{n-1}$ and where $\gamma > \frac{1}{n-1}$ separately. (Middle of page 1182.) And in fact they later also invoke contractions for the first case. Feb 15, 2016 at 15:08

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 $$BR(m^*) = m^*.$$ Suppose we have a non truth-telling equilibrium $m'$. As this is an equilibrium $$BR(m') = m'$$ so $$\forall i\in N: m_i' = \theta_i + \gamma \sum_{j \ne i} (\theta_j-m_j').$$ Rearranging this $$\forall i\in N: 0 = \theta_i - m_i' + \gamma \sum_{j \ne i} (\theta_j-m_j').$$ Adding up all the equations yields \begin{eqnarray*} 0 & = & \sum_i (\theta_i - m_i') + \gamma \sum_i \sum_{j \ne i} (\theta_j-m_j') \\ 0 & = & \sum_i (\theta_i - m_i') + \gamma \cdot (n-1) \cdot \sum_i (\theta_i - m_i') \\ 0 & = & (1 + \gamma \cdot (n-1)) \cdot \sum_i (\theta_i - m_i') . \end{eqnarray*} There are two ways this can hold. Either $\sum_i (\theta_i - m_i') = 0$ or $\gamma = -\frac{1}{n-1}$.
Case 1. Suppose $\sum_i (\theta_i - m_i') = 0$. Then from \begin{eqnarray*} m_i' & = & \theta_i + \gamma \sum_{j \ne i} (\theta_j-m_j') \\ m_i' - \theta_i & = & \gamma \sum_{j \ne i} (\theta_j-m_j') \\ (m_i' - \theta_i) - \gamma \cdot (m_i' - \theta_i) & = & \gamma \cdot (\theta_i - m_i') + \gamma \sum_{j \ne i} (\theta_j-m_j') \\ (1 - \gamma) \cdot (m_i' - \theta_i) & = & \gamma \sum_{j} (\theta_j-m_j') = 0. \end{eqnarray*} In the truth-telling equilibrium this is true for all $i \in N$. If $\gamma = 1$ non truth-telling equilibria are also possible for any number of players, e.g. $n-1$ players reporting their type $+1$ and one player reporting her type $-n+1$. If $\gamma < 1$ and the equilibrium is non truth-telling then this is not possible.
Case 2. Suppose $\gamma = -\frac{1}{n-1}$.
Let $x_i$ denote $\theta_i - m_i'$, so the equation system $$\forall i\in N: 0 = \theta_i - m_i' + \gamma \sum_{j \ne i} (\theta_j-m_j')$$ becomes $$\forall i\in N: 0 = x_i + \gamma \sum_{j \ne i} x_j.$$ For $\gamma = -\frac{1}{n-1}$ the matrix representation of this equation system is invertible so there is only a single solution, and $\forall i: x_i = 0$ is clearly a solution. This means that the only equilibrium is the truth-telling equilibrium.
• @Shane Having thought about this it is as you point out true that $BR$ is not a contraction if $\gamma > \frac{1}{n-1}$. I can offer a linear algebra based proof instead. Jan 30, 2016 at 10:36