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Third price auction: the winner is the highest bidder but this time instead of paying the second highest bid, he would pay the third highest bid. -assume there are at least 3 bidders. -

Average price auction: the winner is the highest bidder but pays the average of his and the second highest bid.

For these,

How to show whether truthtelling is a dominant strategy or not? And how to show whether truthtelling is a nash equlibrium?

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  • $\begingroup$ Somehow, I feel, there is not enough context. What are players' payoffs? $\endgroup$
    – mathse
    Commented May 12, 2014 at 21:46
  • $\begingroup$ Not given :( Dear @mathse $\endgroup$
    – 1190
    Commented May 12, 2014 at 21:49
  • $\begingroup$ Then, I can't answer it. Maybe this makes sense to someone else, but I don't know. $\endgroup$
    – mathse
    Commented May 12, 2014 at 21:50
  • $\begingroup$ Okay thank you so much:) @mathse $\endgroup$
    – 1190
    Commented May 12, 2014 at 21:53

2 Answers 2

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Truth-telling is not a dominant strategy with the third-price auction. Suppose I value an item at £100 and there are two other bids, £200 and £10. I should bid £205 and pay £10 for the item, even though this is more than my private valuation.

Similarly, truth-telling is not dominant in the average-price auction. Suppose I value the item at $100 and the bid is £10. If I bid my true value I will pay £55, but if I bid £12 I will pay only £11.

Both examples show that truth-telling is not a Nash equilibrium, because you could suppose that the other bids are at their truthful values.

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The logic is the same as the second price auction. Suppose you value the object at $v$ and the two highest other bids are $a,b$ with $a \gt b$. If your value is greater than $a$, bidding above your value doesn't hurt, but it doesn't change anything. You get the object and pay $b$, making a profit of $v-b$. If you bid low, you might drop below $a$ and forego the profit. If your value is between $a$ and $b$, by bidding high you might get the object, paying $b$ and making a profit (at the expense of the $a$ bidder). As long as you value the object more than the second highest other bid, you should bid an arbitrarily large amount. This clearly does not encourage truthful bidding.

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