Introduction to Game Theory
Chiara Mocenni
Game Theory
• Strategic interdependence
• players are independent decision makers with sometimes opposite goals
• each player payoff depends on his own and on the other player’s decisions
• Rational players
• the game objective are well defined
• all players are equally intelligent
• all players choose the strategy that maximizes his payoff
Elements of the Game
•
A set of players
•
For each player, a set of alternatives
•
For each player, the ability to order the alternatives
on the basis of the player’s preferences
Who introduced Game Theory
•
Oligopoly and duopoly: Cournot (1838) e Bertrand (1883)
•
Looking for a general theory to solve many
economic questions in the framework of decision theory: Von Neumann-Morgenstern (1944):
•
Nash equilibrium: Nash (1950)
Economic applications
• Theory of oligopoly (market form wherein a market or industry is dominated by a small number of large sellers, the oligopolists)
• Microeconomics (behavior of individuals and firms in making decisions regarding the allocation of scarce resources and the interactions among these individuals and firms)
• Macroeconomics (performance, structure, behavior, and decision-making of an economy as a whole. This includes regional, national, and global economies)
• Economic Policy (setting levels of taxation, government budgets, the money supply and interest rates as well as the labor
market, national ownership, and many other areas of government interventions into the economy)
A classical example: the prisoner’s dilemma game
• Two suspects have been arrested in a major crime and placed in separate isolation cells. Both care much more about their personal freedom than about the welfare of their accomplice.
• A clever prosecutor makes the following offer to each. “You may choose to confess or remain silent”.
• If you confess and your accomplice remains silent I will drop all charges against you (freedom) and use your testimony to ensure that your accomplice does
serious time (4 years in jail). Likewise, if your accomplice confesses while you remain silent, he will go free (freedom) while you do the time (4 years in jail).
• If you both confess I get two convictions, but I'll see to it that you both get early parole (3 years in jail each).
• If you both remain silent, I'll have to settle for token sentences on firearms possession charges (1 year in jail each).
The bi-matrix
Confess Silent
Confess
3 , 3 0 , 4
Silent
4 , 0 1 , 1
Suspect 1
Suspect 2 G2
G1
3 , 3
1 , 1
The battle of sex
•
A couple (husband and wife) decide to hang out
•
They want to do it together
•
But… the husband would rather go to watch a
football match, while tha wife would rather go to the
cinema
Scenario 1
•
Both of them assign a payoff of 2 if they will be able to stay together
•
Both of them assign a payoff of 1 if they are able to go to the preferred place: the husband to the
soccer match and the wife to the cinema
Cinema Soccer
Cinema
2 , 3 0 , 0
Soccer
1 , 1 3 , 2
Husband
Wife
2 , 3
3 , 2
The bi-matrix
Scenario 2
•
Both of them assign a payoff of 1 if they will be able to stay together
•
Both of them assign a payoff of 2 if they are able to go to the preferred place: the husband to the
soccer match and the wife to the cinema
Cinema Soccer
Cinema
1 , 3 0 , 0
Soccer
2 , 2 3 , 1
Husband
Wife
2 , 2
The couple splits!
The bi-matrix
The game Stag - Hunt
• Two individuals go out on a hunt.
• Each can individually choose to hunt a stag or hunt a hare.
• Each player must choose an action without knowing the choice of the other.
• If an individual hunts a stag, they must have the cooperation of their partner in order to succeed.
• An individual can get a hare by themself, but a hare is worth less than a stag.
Stag Hare
Stag
2 , 2 0 , 1
Hare
1 , 0 1 , 1
Hunter 1
Hunter 2
2 , 2
1 , 1
The bi-matrix
The chicken game
•
Two drivers drive towards each other on a collision course
•
One must swerve, or both may die in the crash
•
But if one driver swerves and the other does not, the one who swerved will be called a "chicken", meaning a coward
•
Inspired by the movie “Gioventu’ bruciata” - “Rebel
Without a Cause” (1955)
Straight Swerve
Straight
-10,-10 1 , -1
Swerve
-1 , 1 0 , 0
Driver 1
Driver 2
1 , -1
-1 , 1
The bi-matrix
Hawk and Dove game
• The Hawk first displays aggression, then escalates into a fight until it either wins or is injured (loses). The Dove first displays aggression, but if faced with major escalation runs for safety. If not faced with such escalation, the Dove attempts to share the resource.
• Given that the resource is given the value V, the damage from losing a fight is given cost C:
• If a Hawk meets a Dove he gets the full resource V to himself
• If a Hawk meets a Hawk – half the time he wins, half the time he loses...so his average outcome is then V/2 minus C/2
• If a Dove meets a Hawk he will back off and get nothing – 0
• If a Dove meets a Dove both share the resource and get V/2
Hawk Dove
Hawk (v-c)/2,(v-c)/2 v , 0
Dove 0 , v v/2 , v/2
Animal 1
Animal 2
v , 0
0 , v
The bi-matrix
The Rock-Scissor-Paper
game
Rock Scissor Paper
Rock
0 , 0 1 , -1 -1 , 1
Scissor
-1 , 1 0 , 0 1 , -1
Paper
