By James F. Gimpel
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Extra info for Algorithms in SNOBOL4
13 (Bayesian First Price Auction) Recall the first price auction: all players state a bid, and the winner is the player with maximum bid, and has to pay his bid value as the price. What are optimal strategies for players in this auction? If the valuations of all players are common knowledge, then the player with maximum valuation would state the second valuation as his bid, and win the auction at the same (or slightly bigger) price as in the second price auction. But how should players bid if they do not know all other players’ valuations?
This game has 2n possible outcomes (for the 2n ways the n players can play), therefore the game in matrix form is exponentially large. To circumvent this, in Chapter 7 we will consider a special class of games called graphical games. The idea is that the P1: SBT 9780521872829main CUNY1061-Nisan 0 521 87282 0 July 5, 2007 exercises 14:12 27 value (or payoff) of a player can depend only on a subset of players. We will define a dependence graph G, whose nodes are the players, and an edge between two players i and j represents the fact that the payoff of player i depends on the strategy of player j or vice versa.
1 and used in the Tragedy of the Commons and the Battle of the Sexes – a solution from which no single player can individually improve his or her welfare by deviating. A strategy vector s ∈ S is said to be a Nash equilibrium if for all players i and each alternate strategy si ∈ Si , we have that ui (si , s−i ) ≥ ui (si , s−i ). In other words, no player i can change his chosen strategy from si to si and thereby improve his payoff, assuming that all other players stick to the strategies they have chosen in s.
Algorithms in SNOBOL4 by James F. Gimpel