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Choose an integer between 0 and 100.

The winner is the guess closest to 70% of the class average as of 11:35am.

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{“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 $s_{-i}$}

Dominance and Rationalizability

Today's Agenda

Dominant and Dominated Strategies

Best Responses

Rationalizability and
Iterated Deletion of Dominated Strategies

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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2

Top

Bottom

Left

Right

2

5

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1

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4

1

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5

5

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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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Which strategies are dominated?

1

2

0

0

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1

1

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1

1

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4

4

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U

M

L

R

D

4

4

,

0

0

,

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?

1

2

C

D

C

D

2

2

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3

0

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1

1

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0

3

,

• 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,

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,

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

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1

1

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0

3

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Prisoners' Dilemma

1

2

Cooperate

Defect

Cooperate

Defect

2

2

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3

0

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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.

The First Strategic Tension:

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

Best Response and Rationalizability

• 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

1

2

1

2

,

4

3

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1

4

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1

1

,

Top

Middle

Left

Center

Bottom

Right

3

0

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2

1

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3

2

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8

0

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8

0

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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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2

1

2

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4

3

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1

4

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1

1

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Top

Middle

Left

Center

Bottom

Right

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8

0

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8

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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
\theta_2 = (\frac{1}{3}, \frac{1}{2}, \frac{1}{6})
u_1(U, \theta_2) =
u_1(M, \theta_2) =
u_1(D, \theta_2) =
BR_1(\theta_2) =
\theta_2 = (\frac{1}{2}, \frac{1}{4}, \frac{1}{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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2

0

0

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1

1

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1

1

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4

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U

M

L

R

D

4

4

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0

0

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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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2

0

0

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1

1

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1

1

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4

4

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U

M

L

R

D

4

4

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0

0

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• 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.

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U

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L

R

D

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Iterated Deletion of Dominated Strategies (a.k.a. “Iterated Dominance")

• Suppose it is common knowledge that all players are rational,
so they will never choose a strictly dominated strategy.
• We can remove from the game any strictly dominated strategies
(by either a pure or mixed strategy).

• Removal may create new dominated strategies, remove them too…

• The set of strategies that’s left at the end of that process
is the set of rationalizable strategies.

• turns out to be independent of the order
in which dominated strategies are removed

Which strategy or strategies is strictly dominated for a player?

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2

1

2

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4

3

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1

4

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1

1

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Top

Middle

Left

Center

Bottom

Right

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3

2

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8

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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.

Keynesian Beauty Contest (70% Game)

Everyone chooses an integer between 0 and 100.

The closest person to 70% of the average guess wins.

0

100

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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2

Stag

Hare

Stag

Hare

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5

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4

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4

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Stag Hunt Game

1

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Stag

Hare

Stag

Hare

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5

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4

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4

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4

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Suppose your payoff is the expected value if you were to play against a random person in the class. Which would you choose?

Conclusions and Next Steps

So far we've mainly talked about what people won't do,
and we have a good predictor of what they will do --
but only if they have a single dominant strategy.

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

By Chris Makler

Econ 51 | 09 | Dominance, Best Response, and Rationalizability

The first steps to understanding strategic behavior: what will you NOT do? What, based on your beliefs about the other players' strategies, might you do?

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