3
$\begingroup$

I am interest in finding the optimal strategy for a single bidder (the BNE), when I have a third-price auction with N bidders and two identical goods. The bidders have iid valuations: U(0,1)

I have been told by my teacher that it is a weakly dominant strategy for the bidders to bid their valuations, but I cannot see why this would be?
(and I know from the literature that it is optimal to bid slightly above your true valuation if only one item is auctioned off, i.e. to bid $b_i=\frac{n-1}{n-2}v_i$)

The utility function for bidder i looks like this:
$ u_i(b_i, b_{-i}) = \left\{ \begin{array}{lr} v_i-\max_{(2)}\{b_{-i}\} & \text{if } b_i>\max_{(1)}\{b_{-i}\}>\max_{(2)}\{b_{-i}\} \\ v_i-\max_{(2)}\{b_{-i}\} & \text{if } \max_{(1)}\{b_{-i}\}>b_i>\max_{(2)}\{b_{-i}\} \\ 0 & \text{if } b_i<\max_{(2)}\{b_{-i}\} \end{array} \right. $
since in this auction, the highest and second highest bidder wins a good each, and both pay the third highest bid, because the bidders gain equal value from either good.

If anyone knows why it supposedly is a weakly dominant strategy to bid your true value in this type of auction, or where I might find a good explanation, I would really appreciate any help I can get!

$\endgroup$
2
  • $\begingroup$ Should we assume that $n \geq 3$? Also: I guess that, if this is indeed dominant, then it should be dominant for any value distribution (not just uniform)? $\endgroup$
    – afreelunch
    Nov 2 '21 at 9:49
  • $\begingroup$ Yes, sorry for not clarifying that I assume $n\geq 3$. And indeed, I would think that it should hold for any continuous distribution with increasing probability. The reason why I asked specifically for case with a uniform value distribution is because that is the distribution I assumed in my thesis. $\endgroup$ Nov 2 '21 at 10:57
2
$\begingroup$

Consider bidder $i$. Let $p = \max_{(2)}\{b_{-i}\}$.

By bidding $v_i$, she wins if $v_i > p$ and not if $v_i < p$ (and indifferent if $x_i = p$).

Suppose, however, she bids $z_i < v_i$.

(i) If $v_i > z_i > p$, she still wins the auction and still gets $v_i - p$.

(ii) If $p > v_i > z_i$, she still loses the auction and still gets $0$.

(iii) However, if $v_i > p > z_i$, she now loses instead of winning.

Thus, bidding less than $v_i$ can never improve $i$'s payoffs.

Can you finish the argument for the case $z_i > v_i$?

$\endgroup$
2
  • $\begingroup$ Thank you! It makes good sense. However, do you happen to know if there is a way to solve this as an equation? Or at least write up the expected utility given that each bidder bids her true value, sinse I am also interested in comparing bidder's expected utility in a third-price auction with her expected utility in a second-price auction? $\endgroup$ Nov 2 '21 at 8:44
  • $\begingroup$ I dont see the benefit of attempting to write this as an equation? This is enough to show this is indeed the equilibrium. As for expected utilities, I would advice you to start a new question instead asking for that! $\endgroup$ Nov 2 '21 at 11:45

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.