Questions about Auctions

Question

I am having a hard time figuring out a problem. In a first price auction with a reserve price R and values of the bidders are U[0,1], how do we find expected revenue given the strategy of both of them is V/2. I am thinking of it like this if v < r or v = r revenue is zero. So what we are really looking at is scenarios where v>r for bidder 1 and v < r for bidder 2 (and vice versa) or we have a situation where we have both bidders v > r and so we'll have to take the max of that. Am I on the right track here? Why is this not bayes-nash equilibrium? I having been trying to figure this out forever. Help!

Answer 1

For your future questions, you might wanna know that it is good practice on MathSE to tag assignment or homework question with the homework tag.

Regarding the strategy being a Bayesian-Nash equilibrium, my comment was misleading. I did not take into account the fact that you have a reservation price $R$. The strategy profile $(V/2,V/2)$ is a Bayesian-Nash equilibrium only for $R =0$.

To convince yourself that $(V/2,V/2)$ is not a Bayesian-Nash equilibrium for other values of $R$, consider the example $R = 1/4$. The strategy profile prescribes that players play $V/2$ whatever the actual realization of $V$. Now ask yourself whether this is really optimal when, for instance, $V$ happens to be $1/3$. In this case, does playing $1/6$ really maximizing your expected utility given that the value of the other bidder is distributed as a $U[0,1]$ and that you know that she will always play $V/2$? Once you get some intuition from this example, try to generalize it to any $R>0$.

As for the expected revenue, you do not need to bother about equilibrium considerations. You are being told that the agents play $(V/2,V/2)$, no matter that it be an equilibrium.

Now I realize my comment was maybe misleading again : the random variable I referred to was not $U[0,1]$ but the random variable you describe, namely the random variable which takes values

• $V_1$ when $V_1 > R$ and $V_2 \leq R$
• $V_2$ when $V_2 > R$ and $V_1 \leq R$
• $\max \{V_1,V_2\}$ when $V_1>R$ and $V_2 >R$
• and $0$ otherwise.

This is really the random variable you want to compute the expected value of.