1 Summary - HEC Paris

HEC Paris. Majeure Economie, 2009-2010. Tristan Tomala.
Introduction to Game Theory.
This document summarizes the course: main definitions and results are given
as well as exercises. Most exercises illustrate important applications of the
theory. Classical textbooks are:
”Introduction to game theory”, by M.J. Osborne.
”A course in game theory”, by M.J. Osborne and A. Rubinstein.
”Game theory”, by D. Fudenberg and J. Tirole.
One can also consult the game theory sections of the following microeconomics textbooks:
”A course in microeconomic theory”, by D. Kreps. (French version, PUF).
”Microeconomic theory”, by Mas-Colell, Whinston and Green.
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Summary
1. What is game theory? What is a game? What is a player? What
does ’strategic’ mean in game theory? A bit of history. Main fields of
application of game theory.
2. Preferences and Choices. Individual and collective choices. Paretooptimality, aggregation.
3. Simultaneous move games. Why simultaneous move games? Classical
examples of games. Dominant and dominated strategies. Nash equilibria. Mixed strategies. Applications in oligopoly theory.
4. Sequential move games. Games of perfect information. Why do we play
chess? Games of imperfect information. Back to simultaneous games.
Repetition. Applications: negociation, Stackelberg competition.
5. Bayesian games. Games with incomplete information. What is the difference between incomplete and imperfect information? Applications:
auction theory, reputation.
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6. Mechanism design. The importance of the rules of the game in economic or social interactions, the revelation principle. Applications:
voting, auctions, partnership, public goods.
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2
Introduction
Some dates.
Pre-history:
• 1838, Cournot.
• 1871, Darwin.
• 1912, Zermelo: a theorem on Chess.
• 1921, Borel.
• 1928, Von Neumann.
History:
• 1944, Games and Economic Behavior by John Von Neumann and Oskar
Morgenstern.
• 1950, John Nash: Nash equilibrium.
• 1953, H.W. Kuhn: Extensive form games.
• 1967, J.C. Harsanyi: Games with incomplete information.
• 1976, Aumann and Shapley: the Folk Theorem.
Nobel Prizes (economics):
• 1994: Nash-Harsanyi-Selten.
• 2005: Aumann-Shelling.
• 2007: Hurwicz-Maskin-Myerson.
Concepts:
• Rational agent.
• Multi-player model: strategic thinking, common knowledge of rationality.
• Normal form games: reduction to simultaneous games.
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• Extensive games: modelling of dynamic strategies.
Applications:
• Biology, Computer Science.
• Political Science, Voting.
• Economics: Oligopoly, auctions, negociation, industrial organization,
economics of information, contracts.
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Preferences and choices
3.1
Individual preferences
Let X be a set of outcomes.
Definition 3.1
• A preference relation on X is a relation which is complete and transitive.
• A preference relation is represented by the utility function u : X → R
if,
x preferred to y iff u(x) ≥ u(y).
Properties.
• A preference relation on a finite (countable) set is always represented
by a utility function.
• The utility function is not unique (increasing transformation).
• Some preferences are not representable (eg. lexicographic preference
on [0, 1] × [0, 1]).
Theorem 3.2 Let X = Rn . The preference is continuous if the sets {y :
y x} and {y : y ≺ x} are open. Then it is representable by a continuous
utility function.
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Choices Let X be a set of outcomes. A choice function over X is a mapping
c which associates to every non-empty subset A of X another subset c(A)
such that ∅ 6= c(A) ⊆ A. c(A) are the choices of the agent when she is given
the option to choose an element of A.
A preference relation induces a choice function: c(A) is the set of elements
in A which are maximal for the relation among A (these are the best options
in A according to the preference). In terms of utility, these are the options
that maximize the utility over A.
Conversely, one can derive a preference relation from a choice function as
follows: x is preferred to y iff x ∈ c({x, y}).
Exercise 1
Prove that a preference relation is complete and transitive iff the associated
choice function satisfies the following two conditions:
(α) x ∈ B ⊂ A, x ∈ c(A) =⇒ x ∈ c(B).
(β) x, y ∈ c(B), B ⊂ A, y ∈ c(A) =⇒ x ∈ c(A).
3.2
Collective choices
The question here is how to define a good outcome in a multi-agent context.
Let X be a set of outcomes and N be a set of players. Let ui : X → R be a
utility function for player i (or a preference relation).
• An outcome x is Pareto-dominated if there exists an outcome y such
that for each player i, ui (y) ≥ ui (x) with a strict inequality for at least
one player.
• An outcome is Pareto-optimal if it is not Pareto-dominated.
Remarks. Pareto optimality says nothing on equality or equity. Several
Pareto-optima may exist. Pareto-optima may be graphically represented in
the utility space.
3.3
Aggregation
How to go from individual preferences and choices to collective ones?
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Arrow’s impossibility theorem Let X be a finite set of outcomes and
n be a number of players. Let R be the set of preferences over X. An
aggregation rule is a mapping from Rn to R which associates a social preference to each profile of individual prerefences. For a profile of preferences
R = (R1 , . . . , Rn ), let Rs be the associated social preference.
• The aggregation rule is Paretian if, whenever all players prefer x to y,
so does society.
• The aggregation rule satisifies independance of irrelevant alternatives
IIA if the social ranking of x and y depends only on the individual
rankings of these two outcomes.
• The aggregation rule is dictatorial if there is a player such that the social
preference coincides with this player’s preference (for each profile).
Theorem 3.3 The only aggregation rules which are Paretian and satisfy IIA
are the dictatorial ones. (Arrow’s Theorem).
Gibbard-Satterthwaite theorem Instead of aggregating preferences, one
may want simply to implement an outcome. Let X be a finite set of outcomes
and n be a number of players. Let R be the set of preferences over X. A
social choice function is a mapping f : Rn → X.
• f is Paretian if it selects only Pareto-optimal outcomes.
• f is Strategy-proof if for every player i, every preferences Ri , Ri0 of player
i and preferences R−i for the other players, player i of preference Ri
prefers f (Ri , R−i ) to f (Ri0 , R−i ).
• f is Dictatorial if there exists a player i such that f selects the outcome
that i prefers.
Theorem 3.4 If X is finite with at least three elements, the only social
choice functions which are Paretian, strategy-proof and onto are the dictatorial ones. (GS Theorem).
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4
Simultaneous move games
Definition 4.1 A n-player game is given by:
• A set of players N = {1, . . . , n}.
• A set of strategies Si for each player i. S = ×i Si is the set of strategy
profiles.
• A utility or payoff function ui : S → R for each player i.
Each player selects a strategy in her strategy set. Choices are simultaneous.
Payoffs depend on the profile of strategies chosen.
Players are assumed: - to be rational (to maximize their payoff), - to
know the game and that other players are rational (common knowledge of
rationality and of the game).
Classical games.
C
D
C
D
3, 3 0, 3
4, 0 1, 1
F
T
Prisoner’s Dilemma
H
T
F
T
2, 1 0, 0
0, 0 1, 2
Battle of the Sexes
H
T
1, −1 −1, 1
−1, 1 1, −1
S
G
Matching Pennies
S
G
2, 2 1, 3
3, 1 0, 0
Chicken
The Cournot oligopoly model. Auctions (see exercises).
4.1
Dominant/dominated strategies
Definition 4.2
• A strategy si of player i is dominant if for every other strategy ti and
every strategy profile of the other players s−i ,
ui (si , s−i ) ≥ ui (ti , s−i ).
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• A strategy is weakly dominant if it is dominant and there is at least
one strategy profile of the other players s−i such that
ui (si , s−i ) > ui (ti , s−i ).
• A strategy si of player i is strictly dominant if for every other strategy
ti and every strategy profile of the other players s−i ,
ui (si , s−i ) > ui (ti , s−i ).
Properties. Strictly dominant =⇒ weakly dominant =⇒ dominant. There
may not exist a dominant strategy. There cannot be more than one weakly
dominant strategy.
Definition 4.3
• A strategy si of player i is dominated if there exists another strategy ti
such that for every strategy profile of the other players s−i ,
ui (si , s−i ) ≤ ui (ti , s−i ).
• A strategy is weakly dominated if it is dominated and there is at least
one strategy profile of the other players s−i such that
ui (si , s−i ) < ui (ti , s−i ).
• A strategy si of player i is strictly dominated if there exists another
strategy ti such that for every strategy profile of the other players s−i ,
ui (si , s−i ) < ui (ti , s−i ).
Strictly dominated implies weakly dominated implies dominated. There
may not exist a dominated strategy.
Iterated elimination of strictly dominated strategies (IEDS).
• Start with the initial game. Each player deletes her strictly dominated
strategies. Consider the game with the remaining strategies.
• Iterate the deletion as long as at least one player finds a strictly dominated strategy is her strategy set.
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• If only one strategy remains for each player, the game is said to be
solvable by IEDS.
Property. The order of deletion of strictly dominated strategies does not
change the outcome.
WARNING: This is not the case for deletion of weakly dominated strategies.
4.2
Exercises
Exercise 2
In a second-price auction, n-player compete for buying an indivisible object.
The worth of the object is vi for player i, so that her utility is vi − p if she
purchases the object at price p and 0 if she does not purchase it. Each player
submits a sealed bid to the auctioneer. The winner is the player that submits
the highest bids and she pays the highest price among the other players (ties
are broken by the throw of a dice). Show that the bid bi = vi is a weakly
dominant strategy. Show that this game is not solvable by IEDS (consider 2
players, v1 = 1, v2 = 2, and restrict the possible bids to 0, 1, 2, 3).
Exercise 3
The auction setting is a collective choice problem where an outcome is: who
gets the object and who pays what. Does the GS theorem apply to this
problem?
Let us identify a preference with a valuation. Can you find a Strategy proof and Non-dictatorial social choice function which selects a Paretooptimal outcome?
Exercise 4
There are n players. Each player submits a number in [0, 100] in a sealed
enveloppe. The goal is to be as close as possible to half of the average of the
numbers chosen. Solve this game by IEDS.
Exercise 5
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A Cournot game is a n-player oligopoly game where: each competitor decides
the quantity qi she produces (all produce the same good). The
P market price
is a decreasing function P (Q) of the total quantity Q =
i qi . Player i
substract a production cost ci (qi ) from her profit.
A linear Cournot game is such that: P (Q) = (A − Q)+ and ci (q) = Ci q.
Assume n = 2, C1 = C2 (=0 for simplicity) and solve the game by IEDS.
Explain why it is not possible with three players.
Exercise 6
Show that in a simple majority rule voting procedure between two candidates,
voting for one’s favorite candidate is a weakly dominant strategy (assume no
indifferences). Give an example with three candidates where this fails.
Assume now that voters have to choose between m policies which lie on
a uni-dimensional axis (say from Left-wing to Right-wing): each policy is
identified with a precise spot on the axis. Each voter has a favorite policy
and her utility is a decreasing function of the distance from her favorite policy
to the one actually implemented.
The voting procedure is the following: each voter names a policy and the
median policy is implemented (the one such that half voters stand at the
left and half at the right: assume for simplicity that the number of voters is
odd).
Prove that it is a weakly dominant strategy for each voter to name her
favorite policy. Does this property hold if the average policy is implemented?
Relate this exercise to the GS theorem.
4.3
Nash equilibria
Definition 4.4 A Nash equilibrium is a profile of strategies s such that for
each player i and strategy ti ,
ui (si , s−i ) ≥ ui (ti , s−i )
Equivalent formulation.
• si is a Best-Reply to s−i if ui (si , s−i ) ≥ ui (ti , s−i ), ∀ti .
• si is a Nash equilibrium if and only if each player plays a best-reply to
the strategies of her opponents.
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Or,
• There does not exist a player i and a profitable deviation ti for this
player, ui (ti , s−i ) > ui (si , s−i ).
Exercise 7
Consider the 2-player linear Cournot game with zero unit cost. Draw the
best-reply curves. Deduce the Nash equilibrium.
Prove also that if the players alternatively play a best-reply to the strategy
of the opponent, the strategies eventually converge to the Nash equilibrium.
Exercise 8
Price competition. In a Bertrand game, the structure of the market (demand,
costs) is the same as in the Cournot game. Each competitor announces the
price at which she’s selling the good. The firm quoting the lowest price
serves all the demand (in case of equality the market splits equally). Assume
symmetric and linear cost and prove that there is a unique equilibrium to
this game where each firm sells at the marginal cost. What if the marginal
costs are different?
4.4
Mixed strategies
Definition 4.5
• A mixed strategy is a probability distribution over the
set of (pure) strategies.
• The game played in mixed strategies is as follows: each player chooses
a mixed strategy and draws a pure strategy at random. Random draws
are independent across players. The payoff is the expected payoff.
Theorem 4.6 Every game with finite action sets admits at least one equilibrium in mixed strategies. (Nash Theorem).
Proposition 4.7 A profile of mixed strategies is a Nash equilibrium if and
only if every pure strategy of player i that is played with positive probability
is a best-reply to the strategies of the opponent. (Indifference principle).
Remarks:
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• In particular, all strategies played with positive probability yield the
same expected payoff.
• A strictly dominated strategy is not played at equilibrium. IEDS does
not affect the set of equilibria.
• One may iteratively delete strategies which are strictly dominated by
some mixed strategy. Equilibria are not affected.
4.5
exercises
Exercise 9
Draw the best-reply curves and find the equilibria for: the prisonner’s dilemma,
the Battle of the sexes, Matching Pennies. Find the equilibria. Find them
again using the indifference principle.
Exercise 10
Let t ∈ R be a parameter. Consider the game where each player has two
actions a, b. The payoff of the two players are the same and equal to: 0 if
they play different actions, t if they both play a, 1 − t if they both play b.
Write the matrix of this game. Compute the equilibria for each value of t.
Draw the graph of the correspondence between t and the equilibria of the
associated game.
Exercise 11
The minority game. There are three players. Each of them has to choose
between two options A, B. A player gets 1 if no other player chose the
same option as her, and 0 otherwise. Find all mixed equilibria of this game.
(Hint: remark that there exists no equilibrium such that exactly two players
randomize.)
Exercise 12
Two competing firms race for a prize. The one who invests more gets the
prize V > 0. Investment is irreversible and the invested money is lost, no
matter the outcome. In case of equal investment, the prize is equally shared.
Show that there is no equilibrium in pure strategies (the strategy set is [0, V ]).
Can you find an equilibrium in mixed strategies?
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Exercise 13
There is an infinite sequence of players. Each can either stay home (0) or go
to the beach (1). The payoff of a player who stays home is 0. The payoff of
a player who goes to the beach is 1 if the beach is not too crowded (i.e. the
number of persons on the beach is finite) and -1 if the beach is too crowded
(i.e. the number of persons on the beach is infinite). Show that this game
has no equilibrium. First show this for pure strategies. Extend to mixed
strategies using the following: If (Xi )i P
is a sequence of independent binary
random variables, the probability that i Xi = +∞ is either 0 or 1 (Borel
Cantelli’s lemma).
Exercise 14
Two players have to share N euros. Each of them demands an amount
(integer). If the demands are feasible, the sharing is implemented. Otherwise,
the player naming the least amount is served, the other takes the rest. In
case of unfeasible equal demands, the money is equally shared. Draw the
payoff matrix of this game for some values of N . Proceed to IEDS. Find the
values of N for which the game is solvable. Find the Nash equilibria for each
value of N .
Exercise 15
Consider the game of guessing the average. Now, players can name arbitrarily
large numbers. Show that this is not solvable by IEDS. Prove that there is
a unique equilibrium.
Exercise 16
A congestion game. There are 6 persons driving from A to B. One road goes
through C, the other goes through D. There are thus four road segments: AC,
CB, AD, DB (all are one-way!). The travel time on a segment depends on
the number n of drivers on that segment. One has TAC (n) = TDB (n) = 10n,
TCB (n) = TAD (n) = 50+n. Formulate this problem as a 6-player game where
each players seeks to minimize her travel time. Find the Nash equilibium (it
is unique). Does it minimize the total travel time of the six drivers?
A new road segment (one way) is opened from C to D, there is thus a new
road from A to B: A-C-D-B. One has TCD (n) = 10 + n. Find the equilibrium
of this new game. Is the traffic better?
Hint: to find the equilibria, use the symmetries of the road network and
find a strategy profile such that the travel time is the same on each road.
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5
Games of perfect information
Definition 5.1 A game of perfect information is given by:
• A set of players N = {1, . . . , n}.
• A game tree: a tree and a mapping assigning each node to a player.
• For each player, a utility functions on terminal nodes.
The game unfolds as follows. We start at the root of the tree. The player
to whom the root is assigned, chooses a branch. At the node reached, the
player to whom the node is assigned chooses a branch and so on. When a
node with no out-going branch is reached (a terminal node), the game is over
and payoffs are distributed.
Examples: Chess, Checkers, Go...
The definition can be extended to infinite trees: payoffs depend on the sequence of nodes visited (the play of the game).
Definition 5.2 The Normal Form.
• A strategy of player i in a game with perfect information is a mapping
which associates an action to each node of player i (an action is an
out-going branch). Let Si be the set of strategies of player i.
• A strategy profile s induces a unique play of the game, i.e. a sequence
of visited nodes. The payoff ui (s) is the payoff of player i associated
with this play.
• The Normal form (or strategic form) of the game with perfect information is the game with simultaneous moves (Si , ui )i .
Solution concepts for simultaneous games apply to games with perfect information. The following is a refinement of Nash equilibria.
Definition 5.3 Subgame Perfect Equilibria (SPE).
• Given a game with perfect information and a node z of the tree, the
game tree below z defines a subgame.
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• A SPE is a profile of strategies that induces a Nash equilibrium in every
subgame.
Theorem 5.4 Every finite game (finite game tree) with perfect information
has a SPE (in pure strategies).
Backward induction.
• Solve each subgame of depth one. (This is a one-player game!!)
• Replace each subgame of depth one γ by a terminal node with the
equilibrium payoff of γ.
• Iterate until the root is reached.
This algorithm computes the SPE. This also proves the theorem, by induction.
5.1
Exercises
Exercise 17
Stackelberg. Consider a linear Cournot game. The Stackelberg game is played
as follows: player 1 (the Leader) chooses her quantity, player 2 (the Follower)
knowing the choice of player 1, chooses her own quantity. A Stackelberg
equilibium is a SPE of this extensive form.
Compute it for a 2-player symetric linear Cournot model. Study the nplayer case where player 1 moves first, player 2 moves second, player 3 moves
third,...
Compare the Stackelberg equilibirum with the Cournot equilibrium: from
the point of view of the leader and of the follower. What would happen in
the Bertrand model?
Exercise 18
Negociation 1: Ultimatum. Two players have to share a surplus normalized
to 1. Player 1 proposes a sharing (x, 1 − x). If player 2 accepts, the sharing
is implemented and the payoffs are (x, 1 − x). If player 2 refuses, payoffs are
(0, 0). Find the SPE(s) of this game. Discuss the differences between the
continuous and discrete verions (ie. what if cents are not splittable?). Find
also all Nash equilibrium outcomes.
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Exercise 19
Negociation 2: Rubinstein’s model of alternating offers. The description
starts like the ultimatum. Two players have to share a surplus normalized
to 1.
Stage 1. Player 1 proposes a sharing (x1 , 1 − x1 ). If player 2 accepts, the
sharing is implemented and the payoffs are (x1 , 1 − x1 ). If player 2 refuses,
go to stage 2.
Stage 2. The surplus is now of size δ < 1. Player 2 proposes a sharing
(x2 , 1 − x2 ). If player 1 accepts, the sharing is implemented and the payoffs
are (δx2 , δ(1 − x2 )). If player 1 refuses, go to stage 3.
Players alternate proposals and counter-proposals until an acceptance is
recorded. The size of the surplus is multiplied by δ at each stage.
Prove that (1/(1 + δ), δ/(1 + δ)) is a SPE payoff. (Hint: consider the
strategy that consists in offering precisely this share and in refusing any offer
below δ/(1 + δ)). Prove also that it is the unique SPE payoff.
Exercise 20
Negociation 3: negociation with deadline. Consider Rubinstein’s game with
the additional rule: if no acceptance is recorded before stage T , the process
ends and the payoff is 0 for each player. Let (uT , vT ) be the SPE payoff of
this game. The aim of this exercise is to prove that this exists, that it is
indeed unique and to compute it. Define the game G(x, y) with only two
stages and such that if player 1 refuses player 2’s offer at stage 2, the game
ends and the payoffs are (δ 2 x, δ 2 y), where x, y are non-negative parameters
such that x + y ≤ 1.
1. Prove that G(x, y) has a unique SPE payoff denoted F (x, y) (compute
it).
2. Prove that (uT , vT ) = F (uT −2 , vT −2 ).
3. Conclude.
Exercise 21
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Stones are placed on the cells of a n × m board. Player alternately choose
stones. When a player chooses a stone, she discard all stones in the NorthEast corner above this stone. The player taking the last stone (the SouthWest-most) has lost.
Prove that player 1 (the first to move) has a winning strategy. Find this
strategy if n = m or if m = 2. What if the board extends infinitely to the
North and East?
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Games in extensive form
Definition 6.1 A game in extensive form (or game with imperfect information) is give by:
• A set of players, a game tree and payoff functions.
• For each player i, a partition of her nodes into information sets.
Information sets model the information available to a player when she chooses
an action. Two nodes x, x0 are in the same information set when the player
does not know whether she is at x or at x0 . This has no impact on the
unfolding of the game. It only has an impact on the strategies.
• A strategy of player i is a mapping that associates an action to each
information set. (In particular, when two nodes are in the same information set, the player chooses the same action at these nodes.) Let Si
be the set of these strategies.
• The normal form of the extensive form game is (Si , ui )i where ui is
defined as in the previous section.
Each game admits a Normal form and an Extensive form.
• A mixed strategy is a probability distribution over pure strategies.
• A behavior strategy is a mapping that associates to each information
set, a probability distribution over the set of actions available.
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A player has perfect recall if she never forgets an information she once
knew or an action she once took: (1) if the nodes x, y, w, z are assigned
to player i, if x, y are in disjoint information sets, if w is in the subgame
following x and z is in the subgame following y, then w and z are in disjoint
informations sets. (2) If a, a0 are possible choices of player i at node x, the
information sets following x, a are disjoint from those following x, a0 .
Theorem 6.2 In a game with perfect recall, mixed and behavior strategies
are equivalent. (Kuhn’s theorem).
Definition 6.3
tion set.
• A proper subgame is a subgame that cuts no informa-
• A SPE is a strategy profile that induces a Nash equilibrium in every
proper subgame.
Theorem 6.4 Every finite extensive form game admits a SPE in mixed
strategy.
This is a mix of Backward induction and of Nash’s Theorem.
6.1
Exercises
Exercise 22
Consider a symetric linear Cournot game. Study the equilibria of the game
with one leader and two followers. Then with two leaders and one follower.
Exercise 23
Repeated Games. Let G be a finite simultaneous move game. In the repeated
game, G is played at each stage t and the actions profiles are publicly observed. In the T -stage repeated game, G is played T times and the overall
payoff is the average payoff. In the δ-discounted game,
P G is infinitely repeated
and the overall payoff is the discounted average t≥1 (1 − δ)δ t−1 ui (at ).
1. Write the tree of the game (for a 2 × 2 game repeated 2 times).
2. A history is a finite sequence of action profiles. Check that histories
are one-to-one associated with proper sub-games.
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3. Prove that if the Prisoner’s Dilemma is repeated T times, then at equilibrium, each player Defects at every stage.
4. Construct a SPE of the δ-discounted Prisoner’s Dilemma where each
player Cooperates at each stage (for δ large enough).
5. Using the same reasoning, prove that if two players engage in a repeated
and discounted Ultimatum game, there is a SPE for which the sharing
is ( 12 , 12 ) at each stage.
6. Prove that every repeated game admits a (simple) SPE.
Exercise 24
Cheap Talk. In a game with cheap talk, players are allowed to exchange
costless and non-binding messages before playing. A simple model of cheap
talk is the following. Let G be a simultaneous game and M be a set of
messages.
• At the first stage, each player chooses a messages. Choices are simultaneous and the profile of messages is publicly observed.
• The game G is played.
Let Γ be this game and σ be an equilibrium.
1. Prove that if a profile of messages m has positive probability under σ,
the strategies following m form an equilibrium of G.
2. Deduce that the equilibrium payoffs in Γ are convex combinations of
the equilibrium payoffs of G.
3. Assume M = {0, 1}. Let u, v be two equilibrium payoff vectors of G.
Construct an equilibrium of Γ with payoff 12 u + 21 v.
4. Assume M = [0, 1]. Let u(1), . . . , u(n) be various equilibrium payoff
vectors of G and λ(1), . . . , λ(n)
P be probability weights. Construct an
equilibrium of Γ with payoff k λ(k)u(k).
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Hints. Take X, Y independent random variables in {0, 1} and let Z = 1 if
X 6= Y and Z = 0 otherwise. Show that if X is uniform then so is Z.
Take X, Y independent random variables in [0, 1] and let Z = X + Y if
X + Y < 1 and Z = X + Y − 1 otherwise. Show that if X is uniform then
so is Z.
In both cases, consider the distribution of Z conditional on Y = y and
verify that it is uniform.
7
Bayesian Games
Definition 7.1 A Bayesian game is defined by:
• A set of players N = {1, . . . , n}, an action set Ai for each player i.
• A set of types Θi for each player i.
• A belief pi (θ−i | θi ) of player i of type θi on other player’s types. (Given
his type, player i assigns probabilities to the other player’s types.)
• A payoff function ui : Θ × A → R. (The payoff of player i depends on
actions and types.)
The type of a player represents her information about the game that is played.
It also gives her beliefs about other player’s information. Special cases are
often considered:
• Private values: the payoff of player i depends only on actions and on
her own type. This is restrictive: the type may be a partial information
on the common value of an object (eg. a financial asset).
• Common prior: There is a probability P on Θ such that pi (θ−i | θi ) =
P (θ−i | θi ). In this case, the Bayesian game is a game of imperfect
infomation in which: -an extra player (Nature) selects the type profile
according to P ; -player i is informed of the θi component; -players
chooses actions.
Definition 7.2
• A strategy of player i is a mapping that associates to
every type a (mixed) action.
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• A strategy profile is a Bayesian equilibrium if each player i of type θi
plays a best-reply to the strategies of the opponents (expected payoffs
are evaluated with respect to the beliefs).
In the common prior case, a Bayesian equilibrium is simply a Nash equilibrium of the game with imperfect information.
7.1
Applications
Jury;
Signalling;
Choice of standard;
Auctions (first and second price);
Correlated equilibria.
8
Social choice and Mechanism design
An environment is:
• A set of players N .
• A set of outcomes.
• A set of types Θi for each player and a prior probability on Θ.
• Utilities ui (θi , x) that depend on types and outcomes (private values).
The problem: A benevolent designer chooses an outcome. His aim is to
choose an efficient outcome. He faces two problems: (1) How to aggregate
individual preferences, (2) how to induce players to reveal their preferences?
8.1
Mechanism design
Definition 8.1
• A mechanism is a family of strategy sets (Si )i and a
mapping g : ×i Si → X.
• An environment and a mechanism induce a Bayesian game where: players learn their types, -choose strategies, -and the outcome is chosen
according to g.
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• A social choice function f : Θ → X is implementable in dominant
strategies if there exists a mechanism (M, g) and a profile of weakly
dominant strategies (σi∗ )i in the induced Bayesian game, such that f =
g ◦ σ∗.
• A social choice function f : Θ → X is implementable in Bayesian equilibrium if there exists a mechanism (M, g) and a Bayesian equilibrium
(σi∗ )i of the induced Bayesian game, such that f = g ◦ σ ∗ .
A mechanism represents the rules of the game set by the designer to induce
players to choose the outcome collectively. For instance, an auction is just a
way of allocating an indivisible good.
• A mechanism is direct if Si = Θi for each i. (Each player is asked to
announce her type, ie. her preferences to the designer.)
• A direct mechanism is truthful if the truthful strategies (reporting the
true type) form a Bayesian equilibrium (or are weakly dominant).
Theorem 8.2 A social choice function is implementable if and only if it is
implementable by a direct and truthful mechanism. (Revelation Principle)
This result applies both for implementation in weakly dominant strategies or
in Bayesian equilibria. This allows to write necessary and sufficient conditions
on f to be implementable.
Incentive compatibility
• f is implementable in weakly dominant strategies if and only if for
each player i, each pair of types θi , θi0 , each profile of types of the other
players θ−i ,
ui (θi , f (θi , θ−i )) ≥ ui (θi , f (θi0 , θ−i ))
• f is implementable in Bayesian equilibrium if and only if for each player
i, each pair of types θi , θi0 ,
Eθ−i [ui (θi , f (θi , θ−i )) | θi ] ≥ Eθ−i [ui (θi , f (θi0 , θ−i )) | θi ]
Applications:
• Optimal auctions: revenu equivalence theorem.
• Public goods: Vickrey-Clarke-Groves mechanism.
22
T.Tomala. HEC Majeure Economie
Examen virtuel de théorie des jeux. 2h00.
Exercice 1. Soit le jeu à deux joueurs suivant dans lequel le joueur 1 choisit
la ligne et le joueur 2 la colonne.


(3, 3) (1, 4) (6, 2) (1, 2)
 (4, 1) (0, 0) (6, 0) (3, 0) 

G=
 (2, 9) (0, 9) (6, 8) (5, 6) 
(2, 11) (0, 3) (5, 7) (10, 10)
1. Ce jeu est-il résoluble par élimiation itérée de stratégies strictement
dominées ?
2. Déterminer tous les équilibres de Nash (purs et mixtes) de ce jeu.
3. Les équilibres de Nash sont-ils Pareto-optimaux dans ce jeu? Est-ce
toujours le cas?
Exercice 2. Alice et Bob ont le même ordinateur portable. Malheureusement, les deux ordinateurs ont été volés. L’assurance veut leur rembourser au
juste prix et propose la règle suivante. Alice et Bob doivent annoncer chacun
la valeur estimée de leur ordinateur. Les choix sont faits simultanéments.
Soit x la valeur annoncée par Alice et y la valeur annoncée par Bob.
• Si x = y alors chacun reçoit cette somme.
• Si x < y, alors Alice reçoit x + 2 et Bob reçoit x − 2.
• Si x > y, alors Alice reçoit y − 2 et Bob reçoit y + 2.
On suppose que les valeurs annoncées doivent être choisies parmi les nombres entiers compris entre 2 et 6.
1. Ecrire la matrice de ce jeu.
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2. Montrer que (2, 2) est le seul équilibre de Nash (pur ou mixte). Indications: Le montrer d’abord en pur. Montrer ensuite que la stratégie
6 est forcément jouée avec probabilité zéro dans un équilibre mixte.
Conclure en poursuivant ce raisonnement.
3. On suppose maintenant qu’Alice joue avant Bob : Alice choisit x,
l’annonce à Bob, qui choisit alors y.
Résoudre ce jeu par backward induction et comparer avec l’équilibre de
Nash du jeu simultané.
24
Exercice 3. Deux firmes i = 1, 2 se font concurrence sur un même marché.
Chaque firme i doit choisir sa quantité de production qi ∈ R+ , les choix étant
simultanés. Si la quantité totale produite est Q = q1 + q2 , le prix de marché
du bien est fixé à P (Q) = max(1 − Q, 0).
• La firme 1 qui produit la quantité q1 doit payer un coût de production
C1 (q1 ) = q12 .
• La firme 2 qui produit la quantité q2 doit payer un coût de production
C2 (q2 ) = cq22 avec c > 0.
• Le but de chaque firme est de maximiser son bénéfice net,
bénéfice = quantité × prix de marché − coût.
1. Ecrire le jeu sous forme stratégique associé (ensembles de stratégies,
fonctions de paiements).
2. Montrer que pour toute stratégie q2 du joueur 2, le joueur 1 a une
unique meilleure réponse b1 (q2 ) qui vaut:
b1 (q2 ) = max{
1 − q2
, 0}.
4
3. Montrer que pour toute stratégie q1 du joueur 1, le joueur 2 a une
unique meilleure réponse b2 (q1 ) qui vaut:
b2 (q1 ) = max{
1 − q1
, 0}.
2(c + 1)
4. Déterminer les équilibres de Nash de ce jeu (en justifiant la réponse).
Calculer le prix d’équilibre en fonction du paramètre c.
5. Comparer le prix limite quand c tend vers l’infini avec la situation où
le joueur 1 est en situation de monopole sur le marché.
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HEC Majeure Economie 2008. Examen de théorie des jeux.
2h00. Documents et calculatrices autorisés.
Ordinateurs protables interdits. Anglais autorisé.
Exercice 1. Bataille des sexes avec option d’entrée. Soit la version
suivante de la bataille des sexes (BoS), le joueur 1 choisit la ligne et le joueur
2 la colonne.
F
T
F
3, 1
0, 0
T
0, 0
1, 3
BoS
On considère le jeu Γ où,
• Dans une première étape, le joueur 1 peut décider de jouer le jeu BoS
ou de sortir (S), ce choix étant annoncé au joueur 2. Si il sort, les
paiements sont (2, 2).
• Si il décide de jouer, la bataille des sexes BoS est jouée.
1. Le jeu Γ est-il à information parfaite ? Ecrire précisément l’arbre de ce
jeu.
2. Donner la matrice du jeu Γ. On pourra regrouper les stratégies équivalentes
du joueur 1 en une seule.
3. Déterminer tous les équilibres de Nash de Γ, en stratégies pures, puis
en stratégies mixtes. Donner les paiements des joueurs en chaque
équilibre.
4. Déterminer les équilibres de Nash du jeu BoS, en stratégies pures et en
stratégies mixtes. Peut-on éliminer des équilibres de Γ par ”backward
induction” ?
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5. Quel équilibre de Γ survit à l’élimination des stratégies faiblement
dominées ? Expliquer intuitivement cet équilibre.
Exercice 2. Sur l’ajout de stratégies. Soit le jeu à deux joueurs suivant
dans lequel le joueur 1 choisit la ligne et le joueur 2 la colonne.
H
M
B
G D
6,1 2,0
2,1 6,0
3,0 3,1
1. Ce jeu est-il résoluble par élimination de stratégies dominées ?
2. Soit A la stratégie mixte qui joue H et M avec probabilités (1/2, 1/2).
Calculer le paiement espéré du joueur 1 quand il joue A, et ce pour
chaque stratégie du joueur 2. Déterminer la matrice du jeu ou le joueur
1 peut jouer H, M , B ou A.
3. Peut-on résoudre ce nouveau jeu par élimination de stratégies dominées
?
4. Déterminer tous les équilibres, purs et mixtes du jeu de départ.
Exercice 3. Concurrence en quantité avec coût d’entrée. Deux
firmes i = 1, 2 se font concurrence sur un même marché. Chaque firme i doit
choisir sa quantité de production qi ∈ R+ , les choix étant simultanés. Si la
quantité totale produite est Q = q1 + q2 , le prix de marché du bien est fixé à
P (Q) = 10 − Q. Les fonctions de coût sont: C1 (q) = C2 (q) = 4q.
• 1. Donner le jeu associé, préciser les ensembles de stratégies et les
fonctions de paiements.
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• 2. Déterminer les courbes de meilleure réponse, puis les équilibres de
Nash.
• 3. Calculer les bénéfices à l’équilibre. Calculer également le profit
optimal d’une firme en position de monopole.
On suppose maintenant qu’avant de se faire concurrence, les entreprises
doivent payer un coût d’entrée sur le marché. Le jeu se déroule en deux
étapes:
- Etape 1 : chaque entreprise décide d’entrer sur le marché (In) ou de
rester en dehors (Out). Ces choix sont simultanés. Une entreprise qui reste
dehors a un gain de 0. Chaque entreprise qui entre doit payer un coût d’entrée
F.
-Etape 2 : les entreprises qui sont entrées se font concurrence en quantité.
• 4. Résoudre le jeu en Etape 2. Montrer que pour déterminer les
équilibres sous-jeu-parfaits (SPE) du jeu global, on peut se ramener
à un jeu 2 × 2 dont les actions sont In et Out. Préciser la matrice de
ce jeu.
• 5. Déterminer les équilibres (purs et mixtes) de ce jeu lorque : F < 4,
4 < F < 9, F > 9. Commenter rapidement chaque cas.
• 6. On considère maintenant un grand nombre n de firmes identiques.
Pour chaque valeur du coût d’entrée F , déterminer le nombre maximal
de firmes qui entrent sur le marché à l’équilibre.
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