Third and average price auction 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?

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.