Week 1
Normal form games and Dominant strategies
Reading: Osborne: Ch 2, 3, 12
Introduction
A game is a mathematical model of a situation in which:
• there are several economic agents
• they have to make decisions
• the outcome depends on the decision made by all players
Games in normal (or strategic) form
Game set up
A set of players: N 1,2 n
Player i’s set of strategies: Si IEN
Strategy pro le: 5 S 52 sn
Vector of strategies such that ith component of the vector, sᵢ ∈ Sᵢ, is a strategy for player i
The set of all strategy pro les: 5 151,52 salad
sn Sisacartesianproducteg S5 ab cacs.caas.cba.cb.az
discret s
this set represents all possible outcomes of the game sis strategic
The choices of all player bar player i: s 11.23 5262.31
i s sz si sn s s.si continuous
strategies
Payo s
Player i’s payo function assigns a real number to every possible outcome of the game (ie. not a
preference over their own actions) uᵢ: S —> ℝ i e is apreference etc
overoutcomesincaltruism
y y
canbeobjectiveors ubjective outcomes values
As there is uncertainty about the outcome of the game we assume the payo s are von-Neumann
Morgenstern utilities
Players seek to maximise uᵢ(s) conditional on all the information Iᵢ that player i has, E[uᵢ(s) | Iᵢ]
Game in normal form:
f N Si ien Vilien
Each player has to choose a strategy sᵢ ∈ Sᵢ
All players chose their strategies simultaneously and independently of each other
There is no uncertainty
Strict dominance:
Si siES
u si.si u sinsi
ts ies i Js strictlydominatessi
Beliefs about opponents choices
Belief of player i is a probability distribution on 5 i , and is restricted by either
• Opponents acting independently
• Opponents coordinating
Normal form games and Dominant strategies
Reading: Osborne: Ch 2, 3, 12
Introduction
A game is a mathematical model of a situation in which:
• there are several economic agents
• they have to make decisions
• the outcome depends on the decision made by all players
Games in normal (or strategic) form
Game set up
A set of players: N 1,2 n
Player i’s set of strategies: Si IEN
Strategy pro le: 5 S 52 sn
Vector of strategies such that ith component of the vector, sᵢ ∈ Sᵢ, is a strategy for player i
The set of all strategy pro les: 5 151,52 salad
sn Sisacartesianproducteg S5 ab cacs.caas.cba.cb.az
discret s
this set represents all possible outcomes of the game sis strategic
The choices of all player bar player i: s 11.23 5262.31
i s sz si sn s s.si continuous
strategies
Payo s
Player i’s payo function assigns a real number to every possible outcome of the game (ie. not a
preference over their own actions) uᵢ: S —> ℝ i e is apreference etc
overoutcomesincaltruism
y y
canbeobjectiveors ubjective outcomes values
As there is uncertainty about the outcome of the game we assume the payo s are von-Neumann
Morgenstern utilities
Players seek to maximise uᵢ(s) conditional on all the information Iᵢ that player i has, E[uᵢ(s) | Iᵢ]
Game in normal form:
f N Si ien Vilien
Each player has to choose a strategy sᵢ ∈ Sᵢ
All players chose their strategies simultaneously and independently of each other
There is no uncertainty
Strict dominance:
Si siES
u si.si u sinsi
ts ies i Js strictlydominatessi
Beliefs about opponents choices
Belief of player i is a probability distribution on 5 i , and is restricted by either
• Opponents acting independently
• Opponents coordinating
