Strategy and payoff a In a game tree, a strategy for a player is represented by a set of edges A combination of strategies (sets of edges) one for each player, induce one path from the root to a terminal node which determines the payoffs of all players 16
Strategy and payoff ◼ In a game tree, a strategy for a player is represented by a set of edges. ◼ A combination of strategies (sets of edges), one for each player, induce one path from the root to a terminal node, which determines the payoffs of all players 16
Sequential-move matching pennies ■ Player1 s strategies Head Tail ■ Player2 s strategies Hif player 1 plays H, H if player 1 plays T H if player 1 plays H, t if player 1 plays T T if player 1 plays H, H if player 1 plays T T if player 1 plays H, T if player 1 plays T Player 2's strategies are denoted by hh, ht, th and TT, respectively. (n x m)
Sequential-move matching pennies ◼ Player 1’s strategies ➢ Head ➢ Tail ◼ Player 2’s strategies ➢ H if player 1 plays H, H if player 1 plays T ➢ H if player 1 plays H, T if player 1 plays T ➢ T if player 1 plays H, H if player 1 plays T ➢ T if player 1 plays H, T if player 1 plays T Player 2’s strategies are denoted by HH, HT, TH and TT, respectively.(n x m) 17
Sequential-move matching pennies Their payoffs Normal-form representation Player 2 HT TH T Player H 1 1 1 1 1 1 1 1 1 1 1 1 1
Sequential-move matching pennies Player 2 HH HT TH TT Player 1 H -1 , 1 -1 , 1 1 , -1 1 , -1 T 1 , -1 -1 , 1 1 , -1 -1 , 1 18 ◼ Their payoffs ◼ Normal-form representation
Nash equilibrium The set of Nash equilibria in a dynamic game of complete information is the set of Nash equilibria of its normal-form 19
Nash equilibrium ◼ The set of Nash equilibria in a dynamic game of complete information is the set of Nash equilibria of its normal-form. 19
Nash equilibrium in a dynamic game We can also use normal-form to represent a dynamic game The set of Nash equilibria in a dynamic game of complete information is the set of Nash equilibria of its normal-form How to find the Nash equilibria in a dynamic game of complete information >Construct the normal-form of the dynamic game of complete information Find the Nash equilibria in the normal-form
Nash equilibrium in a dynamic game ◼ We can also use normal-form to represent a dynamic game ◼ The set of Nash equilibria in a dynamic game of complete information is the set of Nash equilibria of its normal-form ◼ How to find the Nash equilibria in a dynamic game of complete information ➢ Construct the normal-form of the dynamic game of complete information ➢ Find the Nash equilibria in the normal-form 20