pollev.com/chrismakler
Christopher Makler
Stanford University Department of Economics
Econ 51: Lecture 8
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Left
Right
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Right
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Both OK
Both OK
Crash
Crash
Outcomes
Two bikers approach on an unmarked bike path.
Payoffs
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.
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X
Y
Z
Player 1 makes the first move.
Initial node
Player 1's actions at her decision node
(and decision node)
Twist: Gift X is unwrapped,
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.
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Accept X
Reject X
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X
Y
Z
We represent this by having an information set connecting
player 2's decision nodes
after player 1 chooses Y or Z.
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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.
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X
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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.
A strategy is a complete, contingent plan of action for a player in a game.
This means that every player
must specify what action to take
at every decision node in the game tree!
A strategy space is the set of all strategies available to a player.
Player 1 has a single decision:
which gift to give (X, Y, or Z).
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Accept X
Reject X
Accept Wrapped
Reject Wrapped
Accept Wrapped
Reject Wrapped
Player 2 might have to make one of two decisions: accept or reject gift X,
and accept or reject a wrapped gift.
Let's abbreviate these as A/R and A'/R'.
A
R
A'
R'
A'
R'
Then player 2's strategy space is
Therefore player 1's strategy space is
A strategy profile \(s = (s_1,s_2)\) is a vector showing which strategy from their strategy space is chosen by each player.
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The outcome of this is that gift Z is given and rejected, and both players receive a payoff of 0.
Note: the strategy profile specifies which action is taken at every decision node!
Strategy for player \(i\):
Strategy space for player \(i\):
Strategy profile:
(a complete, contingent plan for how player \(i\) will move)
(set of all possible strategies for player \(i\))
(list of strategies chosen by each player \(i = 1,2,...,n\))
Player 1's Strategy Space:
Player 2's Strategy Space:
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How many strategies does player 1 have
in her strategy space?
Player 1's Strategy Space:
Player 2's Strategy Space:
Player 1's Strategy Space:
Player 3's Strategy Space:
Player 2's Strategy Space:
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
Suppose two firms each simultaneously choose a quantity \(q_i\) to produce.
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.
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\(X\)
\(AA'\)
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\(AR'\)
\(RA'\)
\(RR'\)
\(Y\)
\(Z\)
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\(X\)
\(AA'\)
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\(AR'\)
\(RA'\)
\(RR'\)
\(Y\)
\(Z\)
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\(X\)
\(AA'\)
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\(AR'\)
\(RA'\)
\(RR'\)
\(Y\)
\(Z\)
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\(X\)
\(AA'\)
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\(AR'\)
\(RA'\)
\(RR'\)
\(Y\)
\(Z\)
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\(OA\)
\(I\)
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\(O\)
\(OB\)
\(IA\)
\(IB\)
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\(A\)
\(C\)
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\(D\)
\(B\)
\(A\)
\(C\)
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\(D\)
\(B\)
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Both include the players, strategies, and payoffs.
The extensive form also includes information
on timing and information.
We usually use the normal form for
static (simultaneous-move) games of complete information.
Different extensive forms might have the same normal form.
Equilibria with mixed strategies are sometimes the only equilibrium!
A
B
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Y
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Mixed strategy for player 1:
probability distribution
over {A, B}
Belief for player 1:
probability distribution over {X, Y}
2
\({1 \over 6}\)
\({1 \over 3}\)
\({1 \over 2}\)
\(0\)
Player 1's beliefs
\({1 \over 6} \times 6 + {1 \over 3} \times 3 + {1 \over 2} \times 2 + 0 \times 7\)
\(=3\)
Player 1's expected payoffs from each of their strategies
\(X\)
\(A\)
1
\(B\)
\(C\)
\(D\)
\(Y\)
\(Z\)
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Your expected payoff from playing one of your strategies
is the weighted average of the payoffs, weighted by your beliefs about what the other person is playing
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\({1 \over 6}\)
\({1 \over 3}\)
\({1 \over 2}\)
\(0\)
Player 1's beliefs
\(X\)
\(A\)
1
\(B\)
\(C\)
\(D\)
\(Y\)
\(Z\)
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pollev.com/chrismakler
Given these beliefs, what is player 1's expected payoff from playing Y?
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Your expected payoff from playing one of your strategies
is the weighted average of the payoffs, weighted by your beliefs about what the other person is playing
\({1 \over 6}\)
\({1 \over 3}\)
\({1 \over 2}\)
\(0\)
Player 1's beliefs
\({1 \over 6} \times 6 + {1 \over 3} \times 3 + {1 \over 2} \times 2 + 0 \times 7\)
\(=3\)
\({1 \over 6} \times 12 + {1 \over 3} \times 6 + {1 \over 2} \times 0 + 0 \times 5\)
\(=4\)
\({1 \over 6} \times 6 + {1 \over 3} \times 0 + {1 \over 2} \times 6 + 0 \times 11\)
\(=4\)
Player 1's expected payoffs from each of their strategies
\(X\)
\(A\)
1
\(B\)
\(C\)
\(D\)
\(Y\)
\(Z\)
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\(X\)
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\(D\)
\(Y\)
\(Z\)
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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.
\({1 \over 6}\)
\({1 \over 3}\)
\({1 \over 2}\)
\(0\)
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
Complete Information
Perfect Information
All players, strategies, and payoffs are common knowledge.
(Everyone knows what game they are playing, and all relevant attributes of the other players in the game.)
All players know all events in a sequential game which have previously occurred,
including random events.