Lec 2 Static Games of Complete Information

what is a set of feasible strategoes

Si

= stratgey space

what is a static game of complete information

a game of the form : first the players simultaneously choose actions, then the players receive payoffs that depend on the combination of actions just chosen

the payoff function is common knowledge among all players

What is a strategy profile

s = (s1,s2, ...sn) 2 S = n ∏ i=1 Si

= the product of all strategies of player i

DEF : Normal-form representation of an n-player game

specifies the player’s strategy spaces S1, S2, ,, Sn, and their payoff functions u1, u2, ,,un

the game is denoted by G = {S1, S2, ..Sn; u1, u2, ...un}

s-i

= (s1,s2..si-1,si+1, ...sn) species a strategy for all players but i

S-i

set of all strategy profiles for all players but i. s-i e S-i .

ui(si ,si-):

i’s payoff if he chooses si and the others s-i

strictly dominated strategy

if a specific strategy si gives player i a strictly worse outcome than another strategy s*i regardless of what the other players are choosing, then i should never choose si

Definition: In the normal-form game G the strategy si e Si is strictly dominated by si* e Si , if it holds: ui(si ,s-i) < ui(si* ,s-i) for all s-i e S-i

Nash equilibrium + Def

idea : find a strategy combination such that no player has an incentive to unilaterally deviate from this strategy combination - this strategy combination is self-enforcing

Definition: In the normal-form game G the strategy combination s* = (s* 1 ,s* 2 , ..s*n ) e S is a Nash equilibrium (NE), if for all players i it holds:

ui(s*i ,s*-i ) >= ui (si ,s*-i ) for all si e Si

Note : if a strategy combination s is not a NE, then at least one player wants to deviate from this strategy combination

relationship between NE and strictly dominated strategies

in the normal-form game G, any strategy si* that is part of a NE s*=(s1*, s2*, ,, sn*) survives elimination of strictly dominated strategies.

In the normal-form game G, if for each player only one strategy s*i survives iterated elimination of strictly dominated strategies, then the equilibrium combination s*=(s1*, s2*, Sn*) is the only Nash equilibrium of the game