# Probability in Games

Christopher Makler

Stanford University Department of Economics

Econ 51: Lecture 11

• Mixed strategies and expected payoffs
• Dominance and best response
• Equilibrium in mixed strategies
• Evolutionary game theory
• Games of incomplete information
• Auctions

# Matching Pennies I: Coordination Game

1

2

Tails

Tails

1

1

,

-1

-1

,

1

1

,

-1

-1

,

Each player chooses Heads or Tails.

If they choose the same thing,
they both "win" (get a payoff of 1).

If they choose differently,
they both "lose" (get a payoff of -1).

Circle best responses.
What are the Nash equilibria of this game?

# Matching Pennies II: Zero-Sum Game

1

2

Tails

Tails

1

-1

,

-1

1

,

1

-1

,

-1

1

,

Each player chooses Heads or Tails.

If they choose the same thing,
player 1 "wins" (gets a payoff of 1)
and player 2 "loses" (gets a payoff of -1).

If they choose differently,
they player 1 "loses" (gets a payoff of -1)
and player 1 "wins" (gets a payoff of 1).

Circle best responses.
What are the Nash equilibria of this game?

• Before you play a game, we assume you know that strategies the other player might play.
• Your best choice depends on what you believe about the other player's strategy.
• Could be your belief about a mixed strategy, or about the other player's "type" or some other random state of the world.

# Beliefs

• Suppose you're kicking a penalty shot in a soccer match.
• Strategy space: {L, R}
• You could choose in advance which way to kick the ball;
this is called a "pure strategy."
• Or you might choose
which way to kick the ball randomly, or at the last minute;
this is called a "mixed strategy."

# Mixed Strategy

• Play one element of your strategy space
with probability 1, others with probability 0
• Example: "Play heads" or "play tails"
• This is what we've been looking at so far
(sorry for any confusion on HW)

# Pure Strategy

• Place positive probability on more than one element of your strategy space
• Example: "Play heads with 50% probability, tails with 50% probability"

Equilibria with mixed strategies are sometimes the only equilibrium!

# Beliefs and Mixed Strategies: Two Different Probability Vectors

A

B

X

Y

1

2

5

4

5

0

0

4

4

4

Mixed strategy for player 1:
probability distribution over {A, B}

Belief for player 1:
probability distribution over {X, Y}

2

If the other player is playing a mixed strategy,
is the weighted average of the payoffs

# Expected Payoffs

$${1 \over 6}$$

$${1 \over 3}$$

$${1 \over 2}$$

$$0$$

Player 2's strategy

$${1 \over 6} \times 6 + {1 \over 3} \times 3 + {1 \over 2} \times 2 + 0 \times 7$$

$$=3$$

$${1 \over 6} \times 12 + {1 \over 3} \times 6 + {1 \over 2} \times 0 + 0 \times 5$$

$$=4$$

$${1 \over 6} \times 6 + {1 \over 3} \times 0 + {1 \over 2} \times 6 + 0 \times 11$$

$$=4$$

Player 1's expected payoffs from each of their strategies

$$X$$

$$A$$

1

$$B$$

$$C$$

$$D$$

$$Y$$

$$Z$$

6

6

,

3

6

,

2

8

,

7

0

,

12

6

,

6

3

,

0

2

,

5

0

,

6

0

,

0

9

,

6

8

,

11

4

,

$$X$$

$$A$$

1

2

$$B$$

$$C$$

$$D$$

$$Y$$

$$Z$$

6

6

,

3

6

,

2

8

,

7

0

,

12

6

,

6

3

,

0

2

,

5

0

,

6

0

,

0

9

,

6

8

,

11

4

,

If  you are playing a mixed strategy, and the other player is  playing a pure strategy, your expected payoff is the weighted average given the way you are mixing.

# Expected Payoffs

$${1 \over 6}$$

$${1 \over 3}$$

$${1 \over 2}$$

$$0$$

Player 2's strategy

$${1 \over 6} \times 6 + {1 \over 3} \times 6 + {1 \over 2} \times 8 + 0 \times 0$$

$$=7$$

$${1 \over 6} \times 6 + {1 \over 3} \times 3 + {1 \over 2} \times 2 + 0 \times 0$$

$$=3$$

$${1 \over 6} \times 0 + {1 \over 3} \times 9 + {1 \over 2} \times 8 + 0 \times 4$$

$$=7$$

Player 2's expected payoffs given each of 1's strategies

# Dominance with mixed strategies

1

2

0

0

,

1

1

,

1

1

,

4

4

,

Top

Middle

Left

Right

Bottom

4

4

,

0

0

,

Are any of player 1's strategies
dominated by a pure strategy?

Are any of player 1's strategies
dominated by a mixed strategy?

1

2

0

0

,

1

1

,

1

1

,

4

4

,

Top

Middle

Left

Right

Bottom

4

4

,

0

0

,

Are any of player 1's strategies
dominated by a mixed strategy?

Suppose player 1 plays Top with 50% probability,

Middle with 0% probability,
and Bottom with 50% probability.

Prob. $${1 \over 2}$$

Prob. 0

Prob. $${1 \over 2}$$

$$\mathbb{E}(u)$$

If player 2 plays Left, what is player 1's expected utility?

If player 2 plays Right, what is player 1's expected utility?

This is better than the utility from playing Middle no matter what player 2 does => Middle is dominated by $$({1 \over 2},0,{1 \over 2})$$

We write this mixed strategy $$({1 \over 2},0,{1 \over 2})$$

2

2

1

2

0

0

,

1

1

,

1

1

,

4

4

,

Top

Middle

Left

Right

Bottom

4

4

,

0

0

,

Prob. $${1 \over 2}$$

Prob. 0

Prob. $${1 \over 2}$$

$$\mathbb{E}(u)$$

This is better than the utility from playing Middle no matter what player 2 does => Middle is dominated by $$({1 \over 2},0,{1 \over 2})$$

2

2

Note that Middle is never a best response to Left or Right. There's a formal result in that in games like this, if a strategy is never a best response, it must be dominated;
so if it's not dominated by a pure strategy,
it must be dominated by some mixed strategy.

## Best Response

If two or more pure strategies are best responses given what the other player is doing, then any mixed strategy which puts probability on those strategies (and no others) is also a best response.

2

$${1 \over 6}$$

$${1 \over 3}$$

$${1 \over 2}$$

$$0$$

Player 2's strategy

$${1 \over 6} \times 6 + {1 \over 3} \times 3 + {1 \over 2} \times 2 + 0 \times 7$$

$$=3$$

$${1 \over 6} \times 12 + {1 \over 3} \times 6 + {1 \over 2} \times 0 + 0 \times 5$$

$$=4$$

$${1 \over 6} \times 6 + {1 \over 3} \times 0 + {1 \over 2} \times 6 + 0 \times 11$$

$$=4$$

Player 1's expected payoffs from each of their strategies

$$X$$

$$A$$

1

$$B$$

$$C$$

$$D$$

$$Y$$

$$Z$$

6

6

,

3

6

,

2

8

,

7

0

,

12

6

,

6

3

,

0

2

,

5

0

,

6

0

,

0

9

,

6

8

,

11

4

,

If player 2 is choosing this strategy, player 1's best response is to play either Y or Z.

Therefore, player 1 could also choose to play any mixed strategy $$(0, p, 1-p)$$.

# When is a mixed strategy a best response?

1

2

Tails

Tails

1

-1

,

-1

1

,

1

-1

,

-1

1

,

$$(p)$$

$$(1-p)$$

What is player 1's expected payoff from Heads?

Suppose player 2 is playing a mixed strategy: Heads with probability $$p$$,
and tails with probability $$1-p$$.

What is player 1's expected payoff from Tails?

1 \times p + -1 \times (1-p)
= 2p - 1
-1 \times p + 1 \times (1-p)
= 1 - 2p

For what value of $$p$$ would player 1 be willing to mix?

2p - 1 = 1 - 2p
\Rightarrow p = {1 \over 2}

# When is a mixed strategy a best response?

1

2

Tails

Tails

1

-1

,

-1

1

,

1

-1

,

-1

1

,

$$(p)$$

$$(1-p)$$

For what value of $$p$$ would player 1 be willing to mix?

p = {1 \over 2}

Now suppose player 1 does mix, and plays Heads with probability $$q$$ and Tails with probability $$1 - q$$.

$$(q)$$

$$(1-q)$$

For what value of $$q$$ would player 2 be willing to mix?

\text{By the same logic, }q = {1 \over 2}

## Equilibrium in Mixed Strategies

A mixed strategy profile is a Nash equilibrium if,
given all players' strategies, each player is mixing among strategies which are their best responses
(i.e. between which they are indifferent)

Important: nobody is trying to make the other player(s) indifferent; it's just that in equilibrium they are indifferent.

1

2

Hawk

Dove

Hawk

Dove

-2

-2

,

0

4

,

2

2

,

4

0

,

# Extensive Form

Nodes:

Branches:

Initial node: where the game begins

Decision nodes: where a player makes a choice; specifies player

Terminal nodes: where the game ends; specifies outcome

Individual actions taken by players; try to use unique names for the same action (e.g. "left") taken at different times in the game

Information sets:

Sets of decision nodes at which the decider and branches are the same, and the decider doesn't know for sure where they are.

A "tree" representation of a game.

She chooses to give one of three gifts:
X, Y, or Z.

1

X

Y

Z

Player 1 makes the first move.

Initial node

Player 1's actions at her decision node

(and decision node)

but Gifts Y and Z are wrapped.
(Player 1 knows what they are,
but player 2 does not.)

After each of player 1's moves,
player 2 has the move: she can either accept the gift or reject it.

2

Accept X

Reject X

2

1

X

Y

Z

We represent this by having an information set connecting
player 2's decision nodes
after player 1 chooses Y or Z.

2

2

Player 2's actions

Player 2's decision nodes

Information set

Accept Y

Reject Y

Accept Z

Reject Z

Also: player 2 cannot make her action contingent on Y or Z; her actions must be "accept wrapped" or "reject wrapped"

Accept Wrapped

Reject Wrapped

Accept Wrapped

Reject Wrapped

After player 2 accepts or rejects the gift, the game ends (terminal nodes) and payoffs are realized.

1

0

1

0

2

0

2

0

3

0

–1

0

2

2

1

X

Y

Z

,

,

,

,

,

,

Accept X

Reject X

Accept Wrapped

Reject Wrapped

Accept Wrapped

Reject Wrapped

Terminal Nodes

Player 1's payoffs

Player 2's payoffs

In this game, both players get a payoff of
0 if any gift is rejected,
1 if gift X is accepted, and
2 if gift Y is accepted.

If gift Z is accepted, player 1 gets a payoff of 3, but player 2 gets a payoff of –1.

By Chris Makler

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