Game Theory I:
SimultaneousMove Games
Christopher Makler
Stanford University Department of Economics
Econ 51: Lecture 9
 Motivation: why game theory?
 Notation and setup
 Components of a game
 The normal form
 Optimal choice
 Dominant and dominated strategies
 Best responses
 Equilibrium
 Equilibrium in dominant strategies
 Iterated deletion of strictly dominated strategies
 Best response (Nash) equilibrium
 Multiple equilibria and belief formation
Today's Agenda
 Up until now: agents only (really) interact with "the market" via prices
 In real life, people, firms, countries ("players") interact with each other.
 Our economic lives are interconnected: our wellbeing doesn't depend only on our own actions, but on the actions taken by others
 Questions:
 OPTMIZATION: How do you operate in a world like this?
 EQUILIBRIUM: What is our notion of "equilibrium" in a world like this, and how is it different from competitive equilibrium?
 POLICY: Given how people behave in strategic settings, how can we design "mechanisms" to achieve policy goals?
Motivation
 The branch of economics that studies strategic interactions between economic agents.
 Everyone's payoffs depend on the actions chosen by all agents
 To "play the game," each agent thinks strategically about how the other agents are playing
Game Theory

Industrial organization: situations where a few firms dominate the market,
and each firm's decisions affect others 
Political economy: campaigning, governing, international diplomacy,
provision of public goods  Contract negotiations: incentive structures, credible threats, negotiating over price
 Interpersonal relationships: team dynamics, division of chores within a family
Applications

Tuesday 11/1: Lecture 9 on simultaneousmove games with pure strategies

Thursday 11/3: Lecture 10 on sequential and repeated games

Sunday 11/6: Homework for lectures 9 & 10 due.

Tuesday 11/8: Democracy Day, no class

Thursday 11/10: Lecture 11 on simultaneousmove games with mixed strategies

Tuesday 11/15: Lecture 12 on simultaneousmove games with incomplete information

Tuesday 11/15: Homework for Lectures 11 & 12 due.

Thursday 11/17: Midterm II
Plan for the next three weeks
 Players: who is playing the game?
 Actions: what can the players do at different points in the game?
 Information: what do the players know when they act?
 Outcomes: what happens, as a function of all players' choices?
 Payoffs: what are players' preferences over outcomes?
Components of a Game
 Outcomes: what happens, as a function of all players' choices?
 Payoffs: what are players' preferences over outcomes?
1
2
1
1
,
0
0
,
1
1
,
0
0
,
Left
Right
Left
Right
1
2
Left
Right
Left
Right
Both OK
Both OK
Crash
Crash
Outcomes
Two bikers approach on an unmarked bike path.
Payoffs
Strategies and Strategy Spaces
A strategy is a complete, contingent plan of action for a player in a game.
This is going to take on more meaning when we look at games that take aren't played simultaneously.
A strategy space is the set of all strategies available to a player.
Continuous: like Cournot; each agent chooses (e.g.) a real number (payoff function)
Discrete: each agent chooses one of a finite number of options (payoff matrix)
NormalForm Game
List of players: \(i = 1, 2, ..., n\)
Strategy spaces for each player, \(S_i\)
Payoff functions for each player \(i: u_i(s)\),
where \(s = (s_1, s_2, ..., s_n)\) is a strategy profile
listing each player's chosen strategy.
Example: Two Player Game (Discrete Strategies)
Strategy for player \(i\):
Strategy space for player \(i\):
Strategy profile:
(set of all possible strategies for player \(i\))
(list of strategies chosen by each player \(i\))
\(X\)
\(A\)
1
2
\(B\)
\(C\)
\(D\)
\(Y\)
\(Z\)
1
1
,
1
1
,
0
0
,
0
0
,
2
2
,
0
0
,
2
2
,
0
0
,
3
–1
,
0
0
,
3
–1
,
0
0
,
Payoffs for both players, as a function of what strategies are played
Example: Two Player Game (Continuous Strategies)
Strategy for player \(i\):
Strategy space for player \(i\):
Strategy profile:
(set of all possible strategies for player \(i\))
(list of strategies chosen by each player \(i\))
Payoffs for both players, as a function of what strategies are played
Notation Convention
Optimal Choice
Optimal choice
 What strategy should a player choose?
 It may depend on what the other player is doing, or it may not
 Dominant strategy: best no matter what the other player does
 Dominated strategy: never a good move
 If there is no dominant strategy, your best move will depend on what the other person is doing (best response).
 Players: prisoners being interrogated in separate rooms.
 Strategies: "cooperate" (don't rat out other)
or "defect" (squeal like the little rat you are) 
Payoffs:
 if they both cooperate, the prosecutor doesn't have much to go on, so they each get a light sentence.
 If they defect while the other cooperates, they go free.
 If they both defect, they both go to jail for a long time.
Prisoners' Dilemma
1
2
Cooperate
Defect
Cooperate
Defect
If you believe the other person will defect,
what is your best response?
If you believe the other person will cooperate, what is your best response?
Defect
Defect
2
2
,
3
0
,
1
1
,
0
3
,
Prisoners' Dilemma
1
2
Cooperate
Defect
Cooperate
Defect
If you believe the other person will defect,
what is your best response?
If you believe the other person will cooperate, what is your best response?
Defect
Defect
Because Defect aways results in a
strictly higher payoff than Cooperate, we say that
Defect strictly dominates Cooperate.
2
2
,
3
0
,
1
1
,
0
3
,
Prisoners' Dilemma
1
2
Cooperate
Defect
Cooperate
Defect
2
2
,
3
0
,
1
1
,
0
3
,
(C,C) pareto dominates (D,D)
(D,D) is a dominant strategy equilibrium
The First Strategic Dilemma:
Everyone doing what's best for themselves can lead to a group loss.
Strict vs. Weak Dominance
1
2
Top
Bottom
Left
Right
2
5
,
1
0
,
4
1
,
5
5
,
Right weakly dominates Left.
Top strictly dominates Bottom.
Iterated Dominance
The process of eliminating strategies that are dominated, until no remaining strategies are dominated.
Rationalizable Strategies
The set of strategies that survive iterated dominance.
Which strategy or strategies is strictly dominated for a player?
1
2
1
2
,
4
3
,
1
4
,
1
1
,
Top
Middle
Left
Center
Bottom
Right
3
0
,
2
1
,
3
2
,
8
0
,
8
0
,
Center strictly dominates Right.
If we know that player 2 will never play Right, is any strategy now dominated for player 1?
Bottom strictly dominates Top.
And with that off the board...
Bottom strictly dominates Middle.
Can we eliminate anything else?
Center strictly dominates Left.
Definition: Best Response
In plain English: given what the other player(s) are doing,
a strategy is my "best response"
if there is no other strategy available to me
that would give me a higher payoff.
1
2
1
2
,
4
3
,
1
4
,
1
1
,
Top
Middle
Left
Center
Bottom
Right
3
0
,
2
1
,
3
2
,
8
0
,
8
0
,
How should player 1 best respond to a belief that player 2 will play Left? What about Center or Right?
Believe Left => play Middle
What about player 2?
Believe Top => play Left
Believe Center => play Bottom
Believe Right => play Top or Bottom
Believe Middle => play Center
Believe Bottom => play Center
Equilibrium
pollev.com/chrismakler
Definition: Best Response (Nash) Equilibrium
In plain English: in a Nash Equilibrium, every player is playing a best response to the strategies played by the other players.
In other words: there is no profitable unilateral deviation
given the other players' equilibrium strategies.
Stag Hunt Game
1
2
Stag
Hare
Stag
Hare
5
5
,
4
0
,
4
4
,
0
4
,
Coordination Game
1
2
Top
Bottom
Left
Right
2
1
,
0
0
,
1
2
,
0
0
,
Pareto Coordination Game
1
2
Top
Bottom
Left
Right
2
2
,
0
0
,
1
1
,
0
0
,
Contribution to a Public Good
 You have $12
 You can contribute $1, $2, $3, $4, $5, or $6
 Your payoff is the amount of money you have left multiplied by the average donation in the class.
Econ 51  9  SimultaneousMove Games
By Chris Makler
Econ 51  9  SimultaneousMove Games
Introduction to game theory; dominance and best response; Nash equilibrium
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