Introuction 1
Brief to Game Theory
a nite set of N players each player i N has a nonempty set of actions Ai each player i N derives a utility ui (a) from action a A =
jN
Denition 1. A strategic game consists of
AJ
Denition 2. A Nash equilibrium of a strategic game N, (Ai ), (ui ) is a prole a A of actions with the property that for every player i N we have (a , a ) i i
i
(a , ai ) for all ai Ai . i
Symmetric games: Ai = Aj ; and same utilities; symmetric equilibrium: a = a i j Denition 3. A Bayesian game consists of a nite set of N players a nite set of states and for each player a set of actions Ai a nite set of signals Ti that may be observed by player i and a function i : Ti . a nondegenerate probability measure pi on that reects the prior of player i a utility ui dened on A
Denition 4. A Nash equilibrium of a Bayesian game N, , (Ai ), (Ti ), (i ), (pi ), (ui ) is a Nash equilibrium of the strategic game dened as follows. 1
the set of players is a set of all pairs (i, ti ) for i N and ti Ti . the set of actions for each player (i, ti ) is Ai . the utility of each player (i, ti ) is dened by Von Neumann Morgenstern utility.
Denition 5. The mixed extension of the strategic game N, (Ai ), (ui ) is the strategic game N, (Ai ), (Ui ) in which (Ai ) is the set of probability distributions over Ai and Ui is a Von Neumann Morgenstern utility.
Denition 6. A mixed strategy Nash equilibrium of a strategic game is a Nash equilibrium of its mixed extension.
Proposition 1. Every nite strategic game has a mixed strategy Nash equilibrium.
Denition 7. An extensive form game with perfect information has the following components. a nite set of N players a set H of nite ir innite sequences that satises 1. The empty sequence is a member of H 2. If (ak )k=1..K H (where K may be ) and L < K then (ak )k=1..L H 3. If an innite sequence (ak ) satises (ak )k=1..L H for every 1 positive integer L then (ak ) H 1 A function P that assigns to each nonterminal history a member of N each player i N derives a utility ui (h) from a terminal history h.
2
Denition 8. A strategy of player i N in an extensive game with perfect information N, H, P, (ui ) is a function that assigns an action in A(h) {a : (h, a) H} to each nonterminal history h for which P (h) = i.
Denition 9. A Nash equilibrium of an extensive game with information perfect is a strategy prole s = (s1 , .., sN ) such that for every player i N we have O(s , s ) i i O(s , si ) for every strategy si of player i. i
Denition 10. The subgame of the extensive game with perfect information that follows the histiry h is the extensive game (h) = N, Hh , Ph , (ui,h ) , where Hh is the set of sequences h of actions for which (h, h ) H, Ph is dened by Ph (h ) = P (h, h ), and ui,h = ui,(h,h )
Denition 11. A subgame perfect equilibrium of an extensive game with perfect information is a strategy prole s such that for every player i N and every nonterminal history h for which P (h) = i we have Oh (s , s ) i,h i,h O(s , si ). i,h
for every strategy si of player i in the subgame (h)
Denition 12. An extensive game has the following components a nite set of N players a set H of sequences that satises 1. The empty sequence is a member of H 2. If (ak )k=1..K H (where K may be ) and L < K then (ak )k=1..L H 3. If an innite sequence (ak ) satises (ak )k=1..L H for every 1 positive integer L then (ak ) H 1 3
A function P that assigns to each nonterminal history a member of N {c} (c is chance). A function fc that associates with every history h for which P (h) = c a probability measure fc (|h) on A(h), where each such probability measure is independent of every other such measure. For each player i N the information partition Fi of {h H : P (h) = i} with...