Lecture Notes 3 Wichtige Begriffe

- players are rational (common knowledge)

- information about the game
[players (N), strategy sets (S), payoffs (u)] (also common knowledge)

information is common knowledge if it is known to all the
players, if each player knows that all the players know it, if each player knows that all the players know that all the players know it and so forth ad infinitum

- a rule that predicts the outcome(s)

- F : [N, (S1, . . . , Sn), (u1, . . . , un)] → s∗ is a rule that
defines an equilibrium based on the possible strategy
combinations and the payoff functions of all the
players

strategy A strictly dominates strategy B if

- for every possible strategy profile of the other agents, playing A yields a higher (>) payoff than playing B

(independent of what the others do, playing A always gives a higher payoff)

strategy A weakly dominates the strategy B if

- for every possible strategy profiles of the other agents, playing A yields an at least as high (≥) a payoff as playing B
- for at least one strategy profile of the other
agents, A yields a higher (>) payoff than
playing B

if there is some strategy A that strictly (weakly) dominates B,
then B is called a strictly (weakly) dominated strategy

(Note: if A strictly dominates B, then A weakly dominates B)

a Dominant Strategy Equilibrium of the game
G = (N, (S1, . . . , Sn), (u1, . . . , un))
is a strategy profile (s1, . . . , sn) such that for every player i in N,

si is a dominant strategy of player i

an iterated dominance equilibrium is a strategy
combination found by deleting a dominated strategy
from the strategy set of one of the players,
recalculating to find which remaining strategies are
dominated, deleting one of them and continuing the
process until only one strategy remains for each player

An equilibrium s∗ = (s∗ 1 , . . . , s∗ n ) is a strategy profile consisting of a best strategy for each of the N players in the game

a Nash equilibrium in a game is a list of strategies, one for each player, such that no player can get a better payoff by switching to some other strategy that is available to her while all others adhere
to strategies specified for them in the list

a Nash Equilibrium of the game, G2 = [{1, 2}, {S1, S2}, {u1, u2}],

is a strategy profile (s∗1 , s∗2 ) such that

s∗ 1 ∈ B1(s∗ 2 )   and    s∗2 ∈ B2(s*1 )

a Nash Equilibrium of game G is a strategy profile,

s∗ = (s∗1, s∗2, . . . , s∗i , . . . , s∗n−1, s∗n)
such that
ui (s∗i , s∗−i ) ≥ ui (si , s∗−i ) for all si ∈ Si
for all i ∈ N

a Nash Equilibrium of the game, G , is a strategy profile
s∗ = (s∗1, s∗2, . . . , s∗i , . . . , s∗n−1, s∗n)
such that
s∗i ∈ Bi (s∗−i )
for all i ∈ N

at a minimum, we want for player i,

s∗i to be the best action when
the others are playing s∗−i = (s∗1 , . . . , s∗i−1, s∗i+1, . . . , s∗n )

That is, s∗i is a best response to s∗−i

The best response correspondence of player i in game
G for s−i is the set of strategies that maximize her
payoff, i.e.
Bi (s−i ) = { ̄s ∈ Si such that ui ( ̄s, s−i ) ≥ ui (si , s−i ) for all si ∈ Si }