Frédéric Mirindi

Strategic Considerations & Payoffs

A bidder $i$'s payoff depends on their bid $b_i$ and the highest bid among all other competitors, $B_{-i} = \max_{j \neq i} \{b_j\}$. The payoff is:

$$ \text{Payoff}_i = \begin{cases} v_i - b_i & \text{if } b_i > B_{-i} \text{ (Win)} \\ 0 & \text{if } b_i < B_{-i} \text{ (Lose)} \\ (v_i - b_i) / k & \text{if } b_i = B_{-i} \text{ and } k \text{ bidders tie for the highest bid (Tie)} \end{cases} $$

Bidding $b_i = v_i$ guarantees a zero payoff if winning. Bidding $b_i > v_i$ guarantees a negative payoff if winning. Therefore, any rational bid must satisfy $b_i < v_i$. However, reducing the bid lowers the probability of winning. The optimal strategy involves maximizing the expected payoff.

Equilibrium Bidding Strategy¹

Assuming $n$ risk-neutral bidders with values $v_i \sim F(v)$ independently on $[0, \bar{v}]$, a symmetric Bayesian Nash Equilibrium (BNE) exists where each bidder uses the same bidding function $b^*(v)$. For the specific case where values are drawn from the uniform distribution $U[0, 1]$ (so $\bar{v}=1$ and $F(v)=v$), the equilibrium bid function for a bidder with value $v_i$ is:

$$ b^*(v_i) = E[Y_1 | Y_1 < v_i] = \frac{n-1}{n} v_i $$

Where $Y_1$ is the highest value among the *other* $n-1$ bidders. This formula shows optimal bids involve "shading" below the true value $v_i$. The extent of shading, $v_i - b^*(v_i) = v_i/n$, decreases as the number of competitors $n$ increases.

Dominant Strategy: Truthful Bidding

A remarkable feature of the Vickrey auction is that bidding one's true value ($b_i = v_i$) is a weakly dominant strategy. This means the strategy $b_i = v_i$ performs at least as well as any other strategy $b_i'$, regardless of the bids submitted by other players.

Proof Outline (Why $b_i = v_i$ is optimal)

Let your value be $v_i$. Let $B_{-i} = \max_{j \neq i} \{b_j\}$ be the highest bid among your opponents. We compare the payoff from bidding $b_i = v_i$ to bidding $b_i' \neq v_i$.

1. Case 1: Consider bidding $b_i' > v_i$ (Overbidding).
   • If $B_{-i} \ge b_i'$, you lose with $b_i'$ and would also lose with $b_i=v_i$. Payoff = 0 in both cases.
   • If $v_i \le B_{-i} < b_i'$, bidding $b_i'$ makes you win and pay $B_{-i}$. Your payoff is $v_i - B_{-i} \le 0$. Bidding $b_i=v_i$ would mean you lose (since $B_{-i} \ge v_i$), giving a payoff of 0. Overbidding leads to a non-positive payoff where truthful bidding yields zero.
   • If $B_{-i} < v_i$, bidding $b_i'$ makes you win and pay $B_{-i}$. Payoff is $v_i - B_{-i} > 0$. Bidding $b_i=v_i$ also makes you win and pay $B_{-i}$, yielding the same positive payoff.
   $\implies$ Overbidding ($b_i' > v_i$) is never strictly better than truthful bidding ($b_i = v_i$), and can be strictly worse.
2. Case 2: Consider bidding $b_i' < v_i$ (Underbidding).
   • If $B_{-i} \ge v_i$, you lose with $b_i'$ and would also lose with $b_i=v_i$. Payoff = 0 in both cases.
   • If $b_i' \le B_{-i} < v_i$, bidding $b_i'$ makes you lose. Payoff = 0. Bidding $b_i=v_i$ would mean you win (since $v_i > B_{-i}$) and pay $B_{-i}$, yielding a payoff of $v_i - B_{-i} > 0$. Underbidding leads to zero payoff where truthful bidding yields a positive payoff.
   • If $B_{-i} < b_i'$, you win with $b_i'$ and pay $B_{-i}$. Payoff is $v_i - B_{-i} > 0$. Bidding $b_i=v_i$ also makes you win and pay $B_{-i}$, yielding the same positive payoff.
   $\implies$ Underbidding ($b_i' < v_i$) is never strictly better than truthful bidding ($b_i = v_i$), and can be strictly worse.

Since truthful bidding $b_i = v_i$ yields a payoff at least as good as any other strategy $b_i'$ regardless of others' bids, it is a weakly dominant strategy.