Static Games from a Player's Perspective

Christopher Makler

Stanford University Department of Economics

Econ 51: Lecture 8

i = \text{Player }i
-i = \text{Player(s) other than }i
\text{Example 1: Consider a strategy profile for four players }s = (s_1, s_2, s_3, s_4)
\text{If we consider player 2, then }s_i = s_2 \text{ and } s_{-i}=(s_1, s_3, s_4)
\text{Example 2: }BR_i(\theta_{-i})\text{ is read:}
\text{“Player $i$'s best response to their beliefs about other players' strategies"}
u_i(s_i,s_{-i}) = \text{Player $i$'s utility from playing $s_i$ when others play $s_{-i}$}

Reminder: Notation

Today's Agenda

Dominant and Dominated Strategies

Best Responses

Rationalizability and
Iterated Deletion of Dominated Strategies

Application: Cournot Model

Dominance

One strategy strictly dominates another strategy
if it always yields a strictly higher payoff
no matter what the other players do.

A pure strategy is dominated for a player if
there’s some other strategy that that player could choose
which would give them a higher payoff
no matter what the other players are doing.

One strategy weakly dominates another strategy
if it never yields a strictly lower payoff
no matter what the other players do,
and sometimes yields a strictly higher payoff.  

Right weakly dominates Left.

Top strictly dominates Bottom.

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Bottom

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Which strategies are dominated?

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Which strategies are dominated?

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Which strategies are dominated?

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U

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How to search for a dominated strategy:

1. Look to see if it's dominated by another pure strategy

2. Look for candidate mixed strategies, especially different strategies with alternating large payoffs

3. Remember: you only need to find one strategy that dominates a strategy for strategy to be dominated.

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Suppose you were playing this (symmetric) game as either player. What would you choose?

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  • 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

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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

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Prisoners' Dilemma

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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.

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Prisoners' Dilemma

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Cooperate

Defect

Cooperate

Defect

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(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.

The First Strategic Tension:

Everyone doing what's best for themselves can lead to a group loss.

Domination vs. Best Response

  • Dominated strategy: some other strategy is better
    no matter what you believe the other players will do

  • Best response: the best strategy (or strategies) to play
    given specific beliefs about what the other players will do

Definition: Best Response

\text{Let }\theta_{-i}\text{ be player }i\text{'s beliefs about the strategies}
\text{We say }s_{i}\text{ is a \textbf{best response} given }\theta_{-i}\text{ if}
u_i(s_i,\theta_{-i}) \ge u_i(s'_i,\theta_{-i})
\text{ for every available strategy }s'_i \in S_i

In plain English: given my beliefs about 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.

\text{ player }i\text{'s payoff from }s_i
\text{ player }i\text{'s payoff from }s'_i
\text{being played by all players other than player }i

Golden Balls

Split

Split

Nick

Ibrahim

Steal

Steal

6,800

6,800

13,600

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In this one-shot game, is there a
(weakly) dominated strategy?

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Middle

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How should player 1 best respond to a belief that player 2 will play Left? What about Center or Right?

Believe Left => play Middle

Believe Center => play Bottom

Believe Right => play Top or Bottom

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How should player 1 best respond to a belief that player 2 will play
Left with probability 1/2, and Center and Right with Probability 1/4 each?

Player 1's expected payoffs given \(\theta = ({1 \over 2}, {1 \over 4}, {1 \over 4})\)

1 \times {1 \over 2} + 1 \times {1 \over 4} + 8 \times {1 \over 4} = 2.75
4 \times {1 \over 2} + 1 \times {1 \over 4} + 3 \times {1 \over 4} = 3
3 \times {1 \over 2} + 2 \times {1 \over 4} + 8 \times {1 \over 4} = 4

What is player 1's best response if they believe player 2 will play L?

What is player 1's best response if they believe player 2 will play R?

What is player 1's best response if they believe player 2 will play
L or R with equal probability?

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What is player 1's best response if they believe player 2 will play L with probability q,
and R with probability 1 - q?

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u_1(U) = 0 \times q + 4 \times (1 - q) = 4 - 4q
q
1-q

PLAYER 1'S BELIEFS

u_1(M) = 1 \times q + 1 \times (1 - q) = 1
u_1(D) = 4 \times q + 0 \times (1 - q) = 4q

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u_1(U) = 0 \times q + 4 \times (1 - q) = 4 - 4q
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PLAYER 1'S BELIEFS

u_1(M) = 1 \times q + 1 \times (1 - q) = 1
u_1(D) = 4 \times q + 0 \times (1 - q) = 4q
\text{Belief }(q)
U
\text{Payoff }(u)
D
0
{1 \over 2}
1
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M
  • There is no belief about what Player 2 might do for which M is a best response.
  • Remember: M was also dominated by a mixed strategy over {U, D}.
  • Result: in a finite two-player game, the set of strategies which are never a best response for any belief is also the set of dominated strategies.

Application: Oligopoly

Applications of Game Theory:
Strategic Interactions between Small Numbers of Agents

  • Very realistic - lots of examples, from families to international diplomacy
  • Models of oligopoly
    • 1838: Cournot
    • 1883: Bertrand
    • 1929: Hotelling
    • 1934: Stackelberg
  • Mid-20th century: John Nash formalized and generalized the notion of strategic equilibrium among small numbers of agents

Competition

  • Lots of "small" firms selling basically the same thing (commodity goods)

Oligopoly

  • A few "medium" or "large" firms selling differentiated products
  • Firms face essentially horizontal demand curve
  • Firms face downward sloping demand curve
  • Interdependence:
    each firm's choice
    affects other firms
  • Independence:
    no individual firm's choice affects other firms

Cournot Duopoly

  • Two firms ("duo" in duopoly)
  • Each chooses how much to produce (quantity competition)
  • Market price depends on the total amount produced
  • Each firm faces a residual demand curve 
    based on the other firm's choice

Simple case: linear demand, constant MC, no fixed costs

P(Q) = 14 - Q
c(q) = 2q
\text{No fixed costs }\Rightarrow MC = AC = 2

What is firm 2's best response function?

P(q_1,q_2) = 14 - (q_1+q_2)
c_2(q_2) = 2q_2
\pi_2(q_2) = P(q_1 + q_2)q_2 - c_2(q_2)
= (14 - q_1 - q_2)q_2 - 2q_2
= 14q_2 - q_1q_2 - q_2^2 - 2q_2
\pi'_2(q_2) = 14-q_1 - 2q_2 - 2 = 0
q_2^*(q_1) = 6-\frac{1}{2}q_1
\text{total revenue}
\text{total cost}
\text{marginal revenue}
\text{marginal cost}

2

P

P(q_2|q_1) = 14-q_1 - q_2
MR(Q) = 14-q_1 - 2q_2
MC(q_2) = 2
6-\frac{1}{2}q_1
14-q_1
q_2

"Firm 2's Residual Demand Curve"

Firm 2's "best response function"

q_2^*(q_1) = 6-\frac{1}{2}q_1

Firm 2's "best response function"

q_1
q_2
q_2^*(q_1) = 6 - {1 \over 2}\hat q_1

Conclusions and Next Steps

Today we've looked at games from the perspective of a player:
what is my best response to what everyone else is doing?

Next time: look for Nash Equilibrium 
in which everyone is best responding to everyone else.