Nash Equilibrium

Christopher Makler

Stanford University Department of Economics

Econ 51: Lecture 9

Bonus Homework Points

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  • You are playing against everyone else in the class.
  • If you choose C, you get 4 extra homework regardless of what other people choose
  • If you choose B, you get 5 points times the fraction of the class that chooses B (so if 50% of the class chooses B, you get 2.5 points; if everyone chooses B, you get 5 points)
  • Option A on the PollEv is just to catch people who are pretending to be in class but aren't, and who are a disappointment to everyone who has ever loved them

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  • Part I: Games with a Unique Pure-Strategy Equilibrium

    • ​Discrete actions

    • Continuous actions

  • Part II: Games with Multiple Equilibria

    • ​Coordination games

    • Games of conflict

  • Part III: Mixed strategies

Today's Agenda

Best Response: From Last Time

\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

Best Response under Strategic Certainty

\text{Let }s_{-i}\text{ be the strategies being played by all players other than player }i
\text{We say }s_{i}^*\text{ is a \textbf{best response} to }s_{-i}\text{ if}
u_i(s_i^*,s_{-i}) \ge u_i(s'_i,s_{-i})
\text{ for every available strategy }s'_i \in S_i

In plain English: given the strategies chosen by the other player(s),
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

Possible rationales for strategic certainty:

  • People play this game all the time, and reasonably expect
    the other player to play according to the equilibrium.
  • The players have agreed on a strategy before the game is played;
    as long as no one has an incentive to deviate, it's OK.
  • An outside mediator (society, the law) recommends a strategy profile

Traffic Light

No Traffic Light

Equilibrium

Definition: Best Response (Nash) Equilibrium

\text{A strategy profile }s^* = s_1^*,s_2^*,...,s^*_n\text{ is a \textbf{Nash Equilibrium} if}
u_i(s^*) \ge u_i(s'_i,s^*_{-i})
\text{ for every available strategy }s'_i \in S_i \text{, for all players } i=1,2,...,n

In plain English: in a Nash Equilibrium, every player is playing a best response to the strategies played by the other players.

\text{ player }i\text{'s equilibrium payoff}
\text{ player }i\text{'s payoff from some deviation }s'_i

In other words: there is no profitable unilateral deviation 
given the other players' equilibrium strategies.

I. Unique Equilibria

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Nash equilibrium occurs when every player is choosing strategy which is a
best response to the strategies chosen by the other player(s)

BR_1(L) =
BR_1(C) =
BR_1(R) =
\{M\}
\{B\}
\{T,B\}
BR_2(T) =
BR_2(M) =
BR_1(B) =
\{L\}
\{C\}
\{C\}

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Nash equilibrium occurs when every player is choosing strategy which is a
best response to the strategies chosen by the other player(s)

BR_1(C) =
\{B\}
BR_1(B) =
\{C\}

Because B is a best response to C,
and C is a Best response to B,
the strategy profile (B,C)
is a Nash Equilibrium.

IMPORTANT: THE NASH EQUILIBRIUM IS THE STRATEGY PROFILE (B,C), NOT THE PAYOFFS (2,1)!

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Nash equilibrium occurs when every player is choosing strategy which is a
best response to the strategies chosen by the other player(s)

Positive Definition

In a Nash equilibrium, no player has a profitable unilateral deviation.

Negative Definition

higher payoff than in equilibrium

only they make a different choice

different than equilibrium choice

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Nash equilibrium occurs when every player is choosing strategy which is a
best response to the strategies chosen by the other player(s)

Positive Definition

In a Nash equilibrium, no player has a profitable unilateral deviation.

Negative Definition

higher payoff than in equilibrium

only they make a different choice

different than equilibrium choice

fix \(s_2 = C\)

Possible unilateral deviations for player 1

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}

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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"

Aside: Symmetric Games

  • If you can swap the names of the players and their strategy spaces and payoffs remain the same, the game is a symmetric game.
  • When a symmetric game has a unique equilibrium, you can solve for the strategy which is a best response to itself!
  • BUT: not all symmetric games only have one equilibrium; we'll see a case later with multiple, and where the players choose different things! 

How to Solve for Equilibrium

  • Equilibrium occurs when all players are playing a best response to the other players
  • Payoff matrix: circle best responses, look for a cell with both payoffs circled. Remember that the equilibrium is the strategies not the payoffs!
  • Continuous strategies: the best response functions form a system of equations. Solve for the strategies which are best responses to each other!
  • In a symmetric game with a unique equilibrium, you can solve for a strategy which is a best response to itself!

Multiple Equilibria

A Simple Coordination Game

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Pareto Coordination

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Stag Hunt Game

  • Proposed by philosopher Jean-Jacques Rousseau in his Discourse on Inequality (1755)​

  • Two hunters independently choose whether to hunt a stag or a hare. ​

  • A player who chooses to hunt a hare always gets a payoff of 4 regardless of the other player's choice.​

  • A stag hunt only succeeds if both players do it.​
    In that case, the players divide the stag, giving a payoff of 5 to each.

  • But if only one player hunts a stag, he fails and gets a payoff of zero. ​

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Conflict!

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Where to Apply

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  • Players: two Stanford students applying to summer internships, at Accenture (A) and Bain (B)
  • Each internship will hire at most one Stanford student
  • Accenture pays $8K;
  • Bain pays $10K
  • If they  apply to different internships, they're both hired
  • If they apply to the same internship, they're each hired with probability \({1 \over 2}\) 

Mixed Strategies

Beliefs for Kicker:
probability distribution
over {Defend Left, Defend Right}

Suppose the kicker believed there was a 50% chance the keeper chooses Defend Left.

What is her best response?

Kick Left (because 80 > 65)

What if she believed there was a 90% chance the keeper chooses Defend Left.

What is her best response?

Kick Right (because 80 > 65)

What if she believes that there is an 80% chance the keeper chooses Defend Left?

What is her best response?

Either one!

Suppose the kicker believed there was a 50% chance the keeper chooses Defend Left.

What is her best response?

Kick Left (because 80 > 65)

What if she believed there was a 90% chance the keeper chooses Defend Left.

What is her best response?

Kick Right (because 80 > 65)

What if she believes that there is an 80% chance the keeper chooses Defend Left?

What is her best response?

Either one!

The best response to a belief is the set of strategies 
which yield the highest payoffs, given those beliefs.

If the keeper is playing the mixed strategy

\sigma_2 = (q,1-q)

(defend left with probability \(q\))

Suppose we think of the kicker's strategy as

\sigma_1 = (p,1-p)

(kick left with probability \(p\))

The best response of the kicker is

We can write their best response strategy as

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With what probability \(p\) would the kicker have to choose \(L\) 
for the goalkeeper to play
a mixed strategy?

(p)
(1-p)

Games with Both Pure and Mixed Strategy Equilibria

Where to Apply

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Suppose the other player is applying to A with probability \(p\). For what value of \(p\) would you be indifferent between applying to the two firms? That is, for what value of \(p\) would you be willing to play a mixed strategy?

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Next Time

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