Game Theory
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Game Theory and Mechanism Design
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Game theory to analyze strategic behavior:
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Given a strategic environment (a “game”), and an
assumption about like behavior (e.g. Nash
Equilibrium), we can predict what will happen.
Mechanism design flips the problem around
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How do we set up a strategic environment so that
if players follow a certain type of behavior (e.g.
Nash Equilibrium), we get a desired outcome?
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Game Theory
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A game consists of..
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Players i=1,…,n
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Possible actions for each player: A1,…,An
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Payoff function for each player: ui(a1,…,an)
Game is commonly known to the players.
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Example
C
D
C
2,2
-1 , 4
D
4 , -1
0,0
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Dominant Strategies
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Defn: Action ai is a dominant strategy for
player i for all ai’ and a-I,
ui(ai,a-i)≥ui(ai’,a-i)
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In other words, player i has a dominant
strategy if she has an action that is optimal
regardless of how other players play.
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Example
A
B
A
4,4
0,3
B
3,0
2,2
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Nash Equilibrium
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The action profile a1,…,an is a Nash
Equilibrium if for each player i, and action ai’,
ui(ai,a-i) ≥ ui(ai’,a-i).
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A Nash Equilibrium means that each player
i’s action is optimal given the actions of the
other players.
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Incomplete Information
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What if the player don’t know each others
payoff functions, e.g. in an auction don’t know
the valuations of the other bidders?
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Dominant strategy is still a dominant strategy:
best action regardless of what others do.
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(Bayesian) Nash equilibrium more subtle:
each player uses best reply given correct
probability assessment of what others will do.
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Allocation Problems
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An allocation problem consists of
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Players i=1,…,n
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A set of possible outcomes or allocations X, and
(possibly) payments by each player p1,…,pn.
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Payoff function for each player: vi(x) – pi
Problem in mechanism design is that the
designer may not know the payoff functions!
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“Good” Allocations
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In different settings, we will try to define “good”
allocations or outcomes.
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In auction problems, a “good” allocation might be
one that is Pareto efficient given everyone’s values.
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In matching problems, a “good” allocation might be
one that is “stable” (to be defined in the next class!).
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Mechanisms
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A mechanism consists of
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A set of possible messages for each player: M1,…,Mn
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An allocation rule: x(m1,…,mn)
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A payment rule: p1(m1,…,mn),…, pn(m1,…,mn)
We will look at mechanisms with and without
payments – if no payments, ignore the p’s.
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Examples of Mechanisms
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In auction problems, a type might be a person’s value,
the messages are the bids, and the auction mechanism
says who wins given the bids and what they pay.
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In matching problems, a type might be a person’s
preferences, the messages are statements of
preferences (not necessarily truthful), and the matching
mechanism says who gets matched to whom or what
given the stated preferences.
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Mechanisms and Games
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A mechanism defines a game
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The possible actions are the messages: M1,…,Mn
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The payoffs are a function of the messages
ui(m1,…,mn) = vi(x(m1,…,mn)) - pi(m1,…,mn)
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Sometimes we will focus on direct revelation
mechanisms: Mi is equal to the the set of i’s
possible payoff functions (or equiv. preferences).
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Strategy-proof Mechanisms
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A strategy for player i specifies what message to
send as a function of player i’s type.
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A mechanism is strategy-proof if each player i has a
strategy that is optimal regardless of the strategies
chosen by the other players.
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Sometimes we’ll look at mechanisms that are not
strategy-proof, but have a Nash equilibrium that
generates good outcomes.
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Applying GT to Market Design
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Studying existing markets
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Designing new markets
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Identify the “rules of the game,” the incentives for the
participants, and how they behave. Then try to understand why
the market functions well, or not so well.
Identify the economic problem to be solved, the players and their
incentives and information. Then try to understand what sort of
market rules would lead to a good outcome.
Economic theory provides a conceptual framework, but need
to use data and experiments to test hypotheses, and identify
things models may have missed.
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