Question

3.6. Consider a first-price, sealed-bid auction in which the bid- ders' valuations are independently and uniformly distributed on (0,1). Show that if there are n bidders, then the strategy of bid- ding (n-1)/n times one's valuation is a symmetric Bayesian Nash equilibrium of this auction.

Answer 1

There is a set of players i el = {1, -, n}. They compete for a single prize. Every player submits their bid bi20, i El. The player with higher bid wins the auction. In the case of tie between players with highest bids, everybody wins with probability 1/k. This is the commonly used standard tiebreaking rule. Player i values the prize as vi i el, if she wins the auction. However, player's i valuation can depend on that who is the winner if she loses the auction. Let us denote these valuations as hij l, if player j wins the auction. Notice that hij can be positive, negative or zero. All the information is common knowledge. The player's utility functions in the model without externalities are: UFP = FP_V-b if player i wins the auction lo, if player j i wins the auction in the case of first price scaled bid auction. LAPSV-b "1-6, if player i wins the auction if player j = i wins the auction in the case of all pay scaled bid auction. In order to realize the specification of my thesis research we are going to consider the following utility functions in the case with externalities: Epe um= (Vi-bi if player i wins the auction h, if player j = i wins the auction

in the case of first price sealed bid auction. (Vi-bj. if player i wins the auction h-b, if player j + i wins the auction UAPE SV in the case of all pay sealed bid auction. The first model is adjusted to Funk's model with standard tie-breaking rule. The second model was proposed by B. Klose and D.Kovenock (2013). In the thesis the first price sealed bid auction with interdependent valuations is considered. First, a number of sets of sufficient conditions in which the pure strategy Nash equilibrium exists are found. Then conditions under which mixed strategy equilibria exist are studied. To facilitate studying the problem, a classification of different types of valuations of players is developed and then equilibrium existence conditions are given for each case. In the thesis, we also consider the model where all players know their own valuation and externalities and have some beliefs about the other players' valuations and externalities. Thus, every player observes his or her own valuation and externalities before he/she bids only. The functional form of a symmetric mixed strategy Nash equilibrium is found