博弈论game chap.3

represented as branches emanating from the root.
Decision node：The end of branch, where the subsequent
is made。
Strategic form and Extensive form These two ways of representing a game are equivalent:
The extensive form:
3.3 Dominant Strategy Solution
Definition(p.41). Strategy Sj′strongly dominates all other strategies of player i if the payoff to is strictly greater than the payoff to any other strategy, regardless of which strategy is chosen by the other player(s).
3.4 Case Study: The Strategic Form Of Art Auctions
Sotheby's Parke-Bernet at Rockefeller Center: 3.2.1 Art Auctions: A Description Objects: drawings by Renoir；Lot #264 Registration: you are handed a numbered bidding paddle. Bidding procedure: Once lot #264 comes up, "all you have to do to bid is to raise your paddle and wait for the auctioneer to acknowledge you---higher bids are automatically set by the auctioneer. If nobody tops your bid, then the auctioneer brings the hammer down to close the sale. 3.2.2 Art Auctions: The Strategic Form Players: individuals who carry paddles Strategies: highest bid he is willing to make Outcomes: the bidder who sets up the highest bid will take home the drawing. Payoffs: amount that your last competitor was willing to pay
The symbols for the three components of the strategic form: P={1,2,…,N}; Ai ; πi (i=1,2,…,N)
Example 1: Prisoners' Dilemma (c = confess, nc = not confess)
Calvin c Klein nc 15,0 1,1 c 5,5 nc 0,15
Example 2
Player 1 \Pl S), π2 (H, S) π1 (L, S), π2 (L, S)
High π1 (H, N), π2 (H, N) Low π1 (L, N), π2 (L, N)
3.1.2 Equivalence with the Extensive Form Pictorial representation:
Game Tree
Root：Tree starts from a root，one of the players has to make a
choice.
Branch：The various choices available to this player are
Example 3: Battle of the Sexes (F = football, 0 = opera)
Wife F Husband O 0,0 1,3 F 3,1 O 0,0
The strategic form can be conveniently represented as a matrix of payoffs whenever there are two players in a game.
every extensive form game can be written in strategic form :All we need to do is write down the set of strategies for each player—in the extensive form—and then write down the associated outcomes and payoffs for every vector of strategies. every strategic form game can be written in extensive form :we have to remember is that a strategy is best thought of as a conditional plan of action. Examples:
π i ( s′j , s−i ) > π i ( si , s−i )
∀si
∀s−i
Definition(p.42). A strategy s'j(weakly) dominates another strategy, say s#j , if it does at least as well as against every strategy of the other players, and against some it does strictly better.
3.1 Strategic Form Games
The strategic form of a game is specified by three objects. 1. the list of players in the game 2. the set of strategies available to each player 3. the payoffs associated with any strategy combination (one strategy per player) In the strategic form (norm form), each player simultaneously chooses a strategy, and the combination of strategy chosen by the players determines a payoff for each player.
eg.1 Represent “Prisoners' Dilemma” in extensive form eg.2 Represent game given by Fig. 2.6 in strategic form.
Example Prisoners' Dilemma
The strategic form:
sets the maximum bid at her true valuation for the Renoir is a dominant strategy.
Summary
1. A strategic form game is described by the list of players, the strategies available to each player, and the payoffs to any strategy combination, one strategy for each player. 2. The strategic form can be conveniently represented as a matrix of payoffs whenever there are two players in a game. With more players, a symbolic representation is more convenient. 3. Every extensive form game can be represented in strategic form. Every strategic form game has at least one extensive form representation. 4. A dominant strategy gives higher payoffs than every other strategy regardless of what the other players do. 5. A dominant strategy solution to a game exists when every player has a dominant strategy. 6. An art auction can be modeled as a strategic form game. Bidding truthfully is a dominant strategy solution in that game.
π i ( s′j , s−i ) ≥ π i ( s # i , s−i )
∀s−i
ˆ ˆ π i ( s′j , s−i ) > π i ( s # i , s−i )
ˆ ∃s − i
When every player has a dominant strategy, the game has a dominant strategy solution.