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? 
 A: 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.
A: 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.
