Game Theory

Game TheoryIntroduction to Game Theory and Some Examples

Jordi Massó

International Doctorate in Economic Analysis (IDEA)Universitat Autònoma de Barcelona (UAB)

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Introduction 1 / 73

1.1- Aim of Game Theory

Game Theory studies situations where decisions taken by rationalagents (players) jointly a¤ect the outcome, and players may havedi¤erent preferences on the set of possible outcomes.

Applied to Economics, Political Sciences, Biology, etc.

1.1- Aim of Game Theory

Game Theory studies situations where decisions taken by rationalagents (players) jointly a¤ect the outcome, and players may havedi¤erent preferences on the set of possible outcomes.

Applied to Economics, Political Sciences, Biology, etc.

1.2.- Decision Theory (one agent)

Let A be the set of actions available to an agent.

Let X be the set of outcomes.

Let L(X ) be the set of probability distributions (lotteries) on X .

A probability distribution on X is a mapping p : X �! [0, 1] such that∑x2X

p(x) = 1 (or, in general, an integrable function with the property

thatZx2X

p(x)dx = 1).

Let u : X �! R be the Bernoulli utility function of the agent (onsure outcomes).

Let g : A �! L(X ) be the outcome function that maps eachpossible action taken by the agent into a probability distribution on X(this relationship may be uncertain).

thatZx2X

p(x)dx = 1).

thatZx2X

p(x)dx = 1).

Let L(X ) be the set of probability distributions (lotteries) on X .A probability distribution on X is a mapping p : X �! [0, 1] such that∑x2X

thatZx2X

p(x)dx = 1).

thatZx2X

p(x)dx = 1).

thatZx2X

p(x)dx = 1).

Assume that agent�s preferences � on L(X ) satisfy continuity andthe independence axiom. Then, there exists a utility functionU : L(X ) �! R that represents � and satis�es the expected utilityproperty. Namely,

for all p, p0 2 L(X ), p � p0 if and only if U(p) � U(p0), andfor all p 2 L(X )

U(p) = ∑x2X

p(x) � u(x).

Let H : A �! R be the induced utility function on the set of actions.Namely, for each a 2 A,

H(a) = U(g(a)).

Assume that the feasible set of actions F coincides with A (i.e.,F = A).

for all p, p0 2 L(X ), p � p0 if and only if U(p) � U(p0), and

for all p 2 L(X )U(p) = ∑

x2Xp(x) � u(x).

H(a) = U(g(a)).

U(p) = ∑x2X

p(x) � u(x).

H(a) = U(g(a)).

U(p) = ∑x2X

p(x) � u(x).

H(a) = U(g(a)).

U(p) = ∑x2X

p(x) � u(x).

H(a) = U(g(a)).

One-Agent Decision Problem

Choose a in order to

maxH(a)

s.t. a 2 F .

Solution (if any): a� (or a set of solutions).

To make the setting more realistic (and general), assume that theoutcome function g has the property that the probability distributionon X depends also on some parameter t 2 T . Then,

g : A� T �! L(X ) andH : A� T �! R, where for all (a, t) 2 A� T , H(a, t) = U(g(a, t)).

The feasible set of actions may also depend on t: for each t 2 T ,F (t) � A is the set of feasible actions at t.

maxH(a)

s.t. a 2 F .

maxH(a)

s.t. a 2 F .

maxH(a)

s.t. a 2 F .

maxH(a)

s.t. a 2 F .

g : A� T �! L(X ) and

H : A� T �! R, where for all (a, t) 2 A� T , H(a, t) = U(g(a, t)).

maxH(a)

s.t. a 2 F .

maxH(a)

s.t. a 2 F .

Given t 2 T , choose a in order to

maxH(a, t)

s.t. a 2 F (t).

Solution (if any): a�(t) (or a set of solutions at every t 2 T ).

maxH(a, t)

s.t. a 2 F (t).

maxH(a, t)

s.t. a 2 F (t).

1.3.- Decision Theory (two players): Game Theory

Set of players (agents) I = f1, 2g.

Let A1 be the set of actions of player 1.

A pair (a1, a2) 2 A1 � A2 is a pro�le of actions.

For i = 1, 2, let ui : X �! R be agent i�s utility function on sureoutcomes.

For i = 1, 2, let Ui : L(X ) �! R be agent i�s utility function onprobability distributions on X .

Let g : A1 � A2 �! L(X ) be the outcome function that maps eachpossible pro�le of actions into a probability distribution on X (now, aplayer may not know g(a1, a2) if he does not know the other player�saction).

Let Hi : A1 � A2 �! R be the induced utility function of playeri = 1, 2 on the set of pro�les of actions. Namely, for each(a1, a2) 2 A1 � A2,

Hi (a1, a2) = Ui (g(a1, a2)).

For each pro�le of actions (a1, a2) 2 A1 � A2, let F1(a2) � A1 andF2(a1) � A2 be the feasible sets of actions of player 1 at a2 andplayer 2 at a1, respectively.

Assume that each player does not know the other player�s action(perhaps because they are taken simultaneously). Then, let

ae2 be the action that player 1 expects (or conjectures) player 2 willchoose,

ae1 be the action that player 2 expects (or conjectures) player 1 willchoose.

Hi (a1, a2) = Ui (g(a1, a2)).

Two-players (Non-cooperative) Problems

Player 1 : Given ae2 choose a1 to

maxH1(a1, ae2)

s.t. a1 2 F1(ae2).

Solution(s): br1(ae2).

maxH2(ae1 , a2)

s.t. a2 2 F2(ae1).

(Nash) Equilibrium: (a�1 , a�2) 2 F1(a�1)� F2(a�2) such that

a�1 = br1(a�2) and a

�2 = br2(a

�1). Namely,

Best reply, given the beliefs: a�1 = br1(ae2) and a

�2 = br2(a

Beliefs are correct: ae1 = a�1 and a

e2 = a

maxH1(a1, ae2)

s.t. a1 2 F1(ae2).Solution(s): br1(ae2).

maxH2(ae1 , a2)

s.t. a2 2 F2(ae1).

�2 = br2(a

�1). Namely,

�2 = br2(a

e2 = a

maxH1(a1, ae2)

maxH2(ae1 , a2)

s.t. a2 2 F2(ae1).

�2 = br2(a

�1). Namely,

�2 = br2(a

e2 = a

maxH1(a1, ae2)

maxH2(ae1 , a2)

�2 = br2(a

�1). Namely,

�2 = br2(a

e2 = a

maxH1(a1, ae2)

maxH2(ae1 , a2)

�2 = br2(a

�1). Namely,

�2 = br2(a

e2 = a

maxH1(a1, ae2)

maxH2(ae1 , a2)

�2 = br2(a

�1). Namely,

�2 = br2(a

e2 = a

maxH1(a1, ae2)

maxH2(ae1 , a2)

�2 = br2(a

�1). Namely,

�2 = br2(a

e2 = a

Two-players (cooperative) Problem

Choose a1 and a2 to

maxH1(a1, a2) +H2(a1, a2)

s.t. a1 2 F1(a2) and a2 2 F2(a1).

Solution(s): a1 and a2.

Questions:

How to split the gains from cooperation H1(a1, a2) +H2(a1, a2)?Bargaining?

Design rules that propose how to divide the obtained maximal amounttaking into account the individual contributions.

Positive as well as normative aspects of these rules.

Stability of the rules, depending on their properties.

Choose a1 and a2 to

maxH1(a1, a2) +H2(a1, a2)

s.t. a1 2 F1(a2) and a2 2 F2(a1).

Questions:

Choose a1 and a2 to

maxH1(a1, a2) +H2(a1, a2)

s.t. a1 2 F1(a2) and a2 2 F2(a1).

Questions:

Choose a1 and a2 to

maxH1(a1, a2) +H2(a1, a2)

s.t. a1 2 F1(a2) and a2 2 F2(a1).

Questions:

Choose a1 and a2 to

maxH1(a1, a2) +H2(a1, a2)

s.t. a1 2 F1(a2) and a2 2 F2(a1).

Questions:

Choose a1 and a2 to

maxH1(a1, a2) +H2(a1, a2)

s.t. a1 2 F1(a2) and a2 2 F2(a1).

Questions:

Choose a1 and a2 to

maxH1(a1, a2) +H2(a1, a2)

s.t. a1 2 F1(a2) and a2 2 F2(a1).

Questions:

1.4.- History of Game Theory

Game Theory was born in Princeton in 1944 with the book Theory ofGames and Economic Behavior, written by John von Neumann andOskar Morgenstern. Content:

Non-cooperative games: two-person zero-sum games (absolutelycompetitive settings).Cooperative games: vNM stable sets (di¢ culty of the cooperativeapproach: indeterminacy of the solution).

Previous works:

Cournot, 1838 (Nash equilibrium in quantities).Bertrand, 1883 (Nash equilibrium in prices).Zermelo, 1913 (chess).Edgeworth, 1925 (contract curve=Core).von Neumann, 1928 (Minimax Theorem).Hotteling, 1929 (Nash equilibrium in localizations).Stackelberg, 1934 (subgame perfect equilibrium).

Non-cooperative games: two-person zero-sum games (absolutelycompetitive settings).

Cooperative games: vNM stable sets (di¢ culty of the cooperativeapproach: indeterminacy of the solution).

Previous works:

Cournot, 1838 (Nash equilibrium in quantities).

Bertrand, 1883 (Nash equilibrium in prices).Zermelo, 1913 (chess).Edgeworth, 1925 (contract curve=Core).von Neumann, 1928 (Minimax Theorem).Hotteling, 1929 (Nash equilibrium in localizations).Stackelberg, 1934 (subgame perfect equilibrium).

Previous works:

Cournot, 1838 (Nash equilibrium in quantities).Bertrand, 1883 (Nash equilibrium in prices).

Zermelo, 1913 (chess).Edgeworth, 1925 (contract curve=Core).von Neumann, 1928 (Minimax Theorem).Hotteling, 1929 (Nash equilibrium in localizations).Stackelberg, 1934 (subgame perfect equilibrium).

Previous works:

Cournot, 1838 (Nash equilibrium in quantities).Bertrand, 1883 (Nash equilibrium in prices).Zermelo, 1913 (chess).

Edgeworth, 1925 (contract curve=Core).von Neumann, 1928 (Minimax Theorem).Hotteling, 1929 (Nash equilibrium in localizations).Stackelberg, 1934 (subgame perfect equilibrium).

Previous works:

Cournot, 1838 (Nash equilibrium in quantities).Bertrand, 1883 (Nash equilibrium in prices).Zermelo, 1913 (chess).Edgeworth, 1925 (contract curve=Core).

von Neumann, 1928 (Minimax Theorem).Hotteling, 1929 (Nash equilibrium in localizations).Stackelberg, 1934 (subgame perfect equilibrium).

Previous works:

Cournot, 1838 (Nash equilibrium in quantities).Bertrand, 1883 (Nash equilibrium in prices).Zermelo, 1913 (chess).Edgeworth, 1925 (contract curve=Core).von Neumann, 1928 (Minimax Theorem).

Hotteling, 1929 (Nash equilibrium in localizations).Stackelberg, 1934 (subgame perfect equilibrium).

Previous works:

Cournot, 1838 (Nash equilibrium in quantities).Bertrand, 1883 (Nash equilibrium in prices).Zermelo, 1913 (chess).Edgeworth, 1925 (contract curve=Core).von Neumann, 1928 (Minimax Theorem).Hotteling, 1929 (Nash equilibrium in localizations).

Stackelberg, 1934 (subgame perfect equilibrium).

Previous works:

In 1950 John Nash makes two extremely important contributions1:

Nash equilibrium (non-cooperative solution).

�Equilibrium Points in N�Person Games,�Proceedings of the NationalAcademy of Sciences, USA 36, 48-49 (1950).

Nash bargaining solution (cooperative solution).

�The Bargaining Problem,�Econometrica 18, 155-162 (1950).

Until the middle of the 70�s the contributors to Game Theory weremostly mathematicians.

N A S HNash Aumann Shapley Harsanyi

Selten

1Read the book A Beautiful Mind, written by Sylvia Nasar in 1998.Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Introduction 12 / 73

Selten

In the middle of the 70�s there is a substantial change, due to twomain reasons:

Harsanyi�s contribution to deal with games of incomplete information(some relevant aspects of the game may not be known to all player; forinstance, the other players�utility functions).

Economic Theory starts to have increasing interest to situations thatare not strictly competitive (strategic analysis of interrelated behavioris indispensable). Competitive behavior is like one-agent decisionproblem:

consumer: given p and m chose x to max u(x) s.t. px � m or

�rm: given p choose y to max py � c(y ).

In the 90�s and 00�s Game Theory becomes one of the main tools forthe new Industrial Organization, Information Economics, MechanismDesign, Political Economy, etc.

1.5.- Non-cooperative versus Cooperative Games

There are two di¤erent approaches and families of (mathematical) modelsto study the interaction of rational agents taking decisions with potentiallycon�icting interests (vNM already made this distinction):

Cooperative Game Theory assumes that players have the possibilityto sign binding agreements (and there are external institutions toimplement such agreements).

vNM presented it as a short cut between a theory with 2 players andwith more than 2 players.

The issue with more than 2 players is that coalitions may form and itsmembers may behave coordinately.

What are the coalition structures that are consistent with rationalbehavior in bargaining, arms control, cost imputation?

Cooperative Game Theory:

Two approaches:

Positive (looks for social stability): Core, vNM stable sets, BargainingSet, etc.

Normative (looks for reasonable and �fair�compromises): Shapleyvalue, Nucleolus, Compromise values, etc.

Two family of models:

Transferable utility (TU-games).

Non-transferable utility (NTU-games).

Two approaches:

Non-cooperative Games assumes that binding agreements are notpossible.

Every player looks after his self-interest.

One (positive) approach: Equilibrium analysis. Nash equilibrium plays afundamental role.

An outcome has to have the property that no player, by unilaterallychanging his behavior, could induce a better outcome.

This distinction between cooperative and non-cooperative games isseen however, as being too rigid. There are at least two lines ofresearch trying to close this gap.

Given a particular class of problems, to study them from the twopoints of view: bargaining, matching, cost-sharing, bankruptcy,voting, etc.

Give to one point of view foundations based on the other point ofview. In particular, to give non-cooperative foundations of thecooperative model (cooperative games are short cuts and they are notusing explicitly all the relevant information about the con�ict).

Nash Program.

1.6.- Examples

Chomp (David Gale)

Gale, D. �A curious Nim-type game,�American MathematicalMonthly 81, 1974.

Two players: I = f1, 2g. We identify 1 with x and 2 with o.

A square with n� n subsquares.

Rules:

Starting by player 1, players have to alternatively choose one of the freesubsquares. If a square is chosen, it is eliminated together with allother squares on the north-east of it.

The player that chooses the last square loses.

1.6.- Examples

Chomp (David Gale)

Rules:

1.6.- Examples

Chomp (David Gale)

Rules:

1.6.- Examples

Chomp (David Gale)

Rules:

1.6.- Examples

Chomp (David Gale)

Rules:

1.6.- Examples

Chomp (David Gale)

Rules:

1.6.- Examples

Chomp (David Gale)

Rules:

1.6.- Examples

Chomp (David Gale)

Rules:

1.6.- Examples

Chomp (David Gale): n� n

1.6.- Examples

Chomp (David Gale)

x x x xx x x xx x x x

1.6.- Examples

o o oo o o

1.6.- Examples

o o ox o o o

1.6.- Examples

Chomp (David Gale)

x x x x x xx x x x x xx x x x x xx x x o o ox x x o o o

1.6.- Examples

x x x x x xx x x x x xx x x x x xx x x o o ox x x o o o

o o o o o

1.6.- Examples

x x x x x xx x x x x xx x x x x xx x x o o ox x x o o ox o o o o o

Player 1 loses. Player 2 wins.

1.6.- Examples

Chomp is not an interesting game: Player 1 has a wining strategy,independently of player 2�s behavior.

Chomp is a boring game: it is known how to win.

Winning strategy of player 1:

1.6.- Examples

o x x x x xo x x x x xo x x x x x

x x x x xx x x x x

1.6.- Examples

o x x x x xo x x x x xo x x x x x

x x x x xx x x x x

1.6.- Examples

o x x x x xo x x x x xo x x x x xo x x x x xo x x x x x

1.6.- Examples

o x x x x xo x x x x xo x x x x xo x x x x xo x x x x x

x x x x x

1.6.- Examples

o x x x x xo x x x x xo x x x x xo x x x x xo x x x x xo x x x x x

Player 1 wins.

1.6.- Examples

o x x x x xo x x x x xo x x x x xo x x x x xo x x x x xo x x x x x

Player 1 wins.

1.6.- Examples

Chomp (David Gale): n� (n+ 1).

It is not an interesting game: Player 1 wins (Zermelo�s Theorem(�nite): either 1 can force a win or 2 can force a win). Proof:

Suppose otherwise, player 2 wins. Hypothesis: player 2 has anoptimal strategy to answer every choice of player 1; i.e, player 2 cananswer every possible decision of player 1 and win at the end.

Consider the following strategy of player 1:

Choose (n, n+ 1).Wait for the choice of player 2, say (i , j) 6= (n, n+ 1).Notice that independently of (i , j) 6= (n, n+ 1), the subsquare(n, n+ 1) would have been eliminated anyway, so the choice of(n, n+ 1) is inessential.Now, player 1 can behave accordingly to the winning strategy of player2, after player 1�s choice of (i , j). By hypothesis, there exists suchoptimal strategy.Hence, player 1 wins. �

1.6.- Examples

Chomp (David Gale): n� (n+ 1).It is not an interesting game: Player 1 wins (Zermelo�s Theorem(�nite): either 1 can force a win or 2 can force a win). Proof:

1.6.- Examples

Choose (n, n+ 1).

Wait for the choice of player 2, say (i , j) 6= (n, n+ 1).Notice that independently of (i , j) 6= (n, n+ 1), the subsquare(n, n+ 1) would have been eliminated anyway, so the choice of(n, n+ 1) is inessential.Now, player 1 can behave accordingly to the winning strategy of player2, after player 1�s choice of (i , j). By hypothesis, there exists suchoptimal strategy.Hence, player 1 wins. �

1.6.- Examples

Choose (n, n+ 1).Wait for the choice of player 2, say (i , j) 6= (n, n+ 1).

Notice that independently of (i , j) 6= (n, n+ 1), the subsquare(n, n+ 1) would have been eliminated anyway, so the choice of(n, n+ 1) is inessential.Now, player 1 can behave accordingly to the winning strategy of player2, after player 1�s choice of (i , j). By hypothesis, there exists suchoptimal strategy.Hence, player 1 wins. �

1.6.- Examples

Choose (n, n+ 1).Wait for the choice of player 2, say (i , j) 6= (n, n+ 1).Notice that independently of (i , j) 6= (n, n+ 1), the subsquare(n, n+ 1) would have been eliminated anyway, so the choice of(n, n+ 1) is inessential.

Now, player 1 can behave accordingly to the winning strategy of player2, after player 1�s choice of (i , j). By hypothesis, there exists suchoptimal strategy.Hence, player 1 wins. �

1.6.- Examples

Choose (n, n+ 1).Wait for the choice of player 2, say (i , j) 6= (n, n+ 1).Notice that independently of (i , j) 6= (n, n+ 1), the subsquare(n, n+ 1) would have been eliminated anyway, so the choice of(n, n+ 1) is inessential.Now, player 1 can behave accordingly to the winning strategy of player2, after player 1�s choice of (i , j). By hypothesis, there exists suchoptimal strategy.

Hence, player 1 wins. �

1.6.- Examples

Chomp (David Gale): n� (n+ 1), with n � 11.

However, it is an interesting game since for n � 11 the optimalstrategy of player 1 (which exists) is not known.

Similar reasoning for chess (with the additional outcome of a drawand taking into account that chess is a �nite game since if the piecesare at the same position 4 times, the outcome is a draw) and for alarge class of games as well.

Two properties of Chomp are important for obtaining theseconclusions:

Perfect information.

Finite number of positions (�nite number of ways of playing the game).

1.6.- Examples

Chomp (David Gale): n� (n+ 1), with n � 11.However, it is an interesting game since for n � 11 the optimalstrategy of player 1 (which exists) is not known.

1.6.- Examples

Pirates (Moulin)

Moulin, H. Axioms of Cooperative Decision Making (1st ed.).Cambridge: Cambridge University Press. Econometric SocietyMonographs, 1988.

10 pirates have a treasure with 100 gold coins.

Problem: How to share them?

Answer:

The oldest pirate (pirate 10) makes a proposal, which is voted.If at least half the pirates approves the proposal (#Yes � 5 and whomakes the proposal also votes) the proposal is implemented.Otherwise (#Yes < 5) the oldest pirate, who made the proposal, isthrow it away to the sea where there are many hungry sharks (!!!!).

The second oldest pirate (pirate 9) makes a proposal (the oldest pirateis not around anymore since he is with the sharks) for the remaining 9pirates....

1.6.- Examples

Pirates (Moulin)

Answer:

1.6.- Examples

Pirates (Moulin)

Answer:

1.6.- Examples

Pirates (Moulin)

Answer:

1.6.- Examples

Pirates (Moulin)

Answer:

1.6.- Examples

Pirates (Moulin)

Answer:

The oldest pirate (pirate 10) makes a proposal, which is voted.

If at least half the pirates approves the proposal (#Yes � 5 and whomakes the proposal also votes) the proposal is implemented.Otherwise (#Yes < 5) the oldest pirate, who made the proposal, isthrow it away to the sea where there are many hungry sharks (!!!!).

1.6.- Examples

Pirates (Moulin)

Answer:

The oldest pirate (pirate 10) makes a proposal, which is voted.If at least half the pirates approves the proposal (#Yes � 5 and whomakes the proposal also votes) the proposal is implemented.

Otherwise (#Yes < 5) the oldest pirate, who made the proposal, isthrow it away to the sea where there are many hungry sharks (!!!!).

1.6.- Examples

Pirates (Moulin)

Answer:

1.6.- Examples

Pirates (Moulin)

Answer:

The second oldest pirate (pirate 9) makes a proposal (the oldest pirateis not around anymore since he is with the sharks) for the remaining 9pirates.

1.6.- Examples

Pirates (Moulin)

Answer:

1.6.- Examples

Pirates (Moulin)

Assumption: all pirates are rational (no rum on board) and want �rstto live, and conditional on being alive, to receive the largest numberof coins. Moreover, everybody knows that everybody is rational (inthe above sense), everybody knows that everybody knows thateverybody is rational, everybody knows that everybody knows thateverybody knows that everybody is rational, ...

Question: What is the most likely proposal and who makes it?

1.6.- Examples

Pirates (Moulin)

1.6.- Examples

Pirates (Moulin)

1.6.- Examples

Pirates (Moulin)

Answer: The oldest pirate makes the proposal (from youngest tooldest) (0, 1, 0, 1, 0, 1, 0, 1, 0, 96) and it is approved with the followingvote (N,Y ,N,Y ,N,Y ,N,Y ,N,Y ).

Why?: By backwards induction (importance of the commonknowledge of the rules and the pirates�rationality).

1.6.- Examples

Pirates (Moulin)

1.6.- Examples

Pirates (Moulin)

1.6.- Examples

Pirates (Moulin)

proposal! 1 2 3 4 5 6 7 8 9 10proposer# Votes

12345678910

1.6.- Examples

Pirates (Moulin)

1 1002345678910

1.6.- Examples

Pirates (Moulin)

1 100 Yes2345678910

1.6.- Examples

Pirates (Moulin)

1 1002 0 100345678910

1.6.- Examples

Pirates (Moulin)

1 100 No2 0 100 Yes345678910

1.6.- Examples

Pirates (Moulin)

1 1002 0 1003 1 0 9945678910

1.6.- Examples

Pirates (Moulin)

1 100 Yes2 0 100 No3 1 0 99 Yes45678910

1.6.- Examples

Pirates (Moulin)

1 1002 0 1003 1 0 994 0 1 0 995678910

1.6.- Examples

Pirates (Moulin)

1 100 No2 0 100 Yes3 1 0 99 No4 0 1 0 99 Yes5678910

1.6.- Examples

Pirates (Moulin)

1 1002 0 1003 1 0 994 0 1 0 995 1 0 1 0 98678910

1.6.- Examples

Pirates (Moulin)

1 100 Yes2 0 100 No3 1 0 99 Yes4 0 1 0 99 No5 1 0 1 0 98 Yes678910

1.6.- Examples

Pirates (Moulin)

1 1002 0 1003 1 0 994 0 1 0 995 1 0 1 0 986 0 1 0 1 0 9878910

1.6.- Examples

Pirates (Moulin)

1 100 No2 0 100 Yes3 1 0 99 No4 0 1 0 99 Yes5 1 0 1 0 98 No6 0 1 0 1 0 98 Yes78910

1.6.- Examples

Pirates (Moulin)

1 1002 0 1003 1 0 994 0 1 0 995 1 0 1 0 986 0 1 0 1 0 987 1 0 1 0 1 0 978910

1.6.- Examples

Pirates (Moulin)

1 100 Yes2 0 100 No3 1 0 99 Yes4 0 1 0 99 No5 1 0 1 0 98 Yes6 0 1 0 1 0 98 No7 1 0 1 0 1 0 97 Yes8910

1.6.- Examples

Pirates (Moulin)

1 1002 0 1003 1 0 994 0 1 0 995 1 0 1 0 986 0 1 0 1 0 987 1 0 1 0 1 0 978 0 1 0 1 0 1 0 97910

1.6.- Examples

Pirates (Moulin)

1 100 No2 0 100 Yes3 1 0 99 No4 0 1 0 99 Yes5 1 0 1 0 98 No6 0 1 0 1 0 98 Yes7 1 0 1 0 1 0 97 No8 0 1 0 1 0 1 0 97 Yes910

1.6.- Examples

Pirates (Moulin)

1 1002 0 1003 1 0 994 0 1 0 995 1 0 1 0 986 0 1 0 1 0 987 1 0 1 0 1 0 978 0 1 0 1 0 1 0 979 1 0 1 0 1 0 1 0 9610

1.6.- Examples

Pirates (Moulin)

1 100 Yes2 0 100 No3 1 0 99 Yes4 0 1 0 99 No5 1 0 1 0 98 Yes6 0 1 0 1 0 98 No7 1 0 1 0 1 0 97 Yes8 0 1 0 1 0 1 0 97 No9 1 0 1 0 1 0 1 0 96 Yes10

1.6.- Examples

Pirates (Moulin)

1 1002 0 1003 1 0 994 0 1 0 995 1 0 1 0 986 0 1 0 1 0 987 1 0 1 0 1 0 978 0 1 0 1 0 1 0 979 1 0 1 0 1 0 1 0 9610 0 1 0 1 0 1 0 1 0 96

1.6.- Examples

Pirates (Moulin)

1 100 No2 0 100 Yes3 1 0 99 No4 0 1 0 99 Yes5 1 0 1 0 98 No6 0 1 0 1 0 98 Yes7 1 0 1 0 1 0 97 No8 0 1 0 1 0 1 0 97 Yes9 1 0 1 0 1 0 1 0 96 No10 0 1 0 1 0 1 0 1 0 96 Yes

1.6.- Examples

Matching Pennies

Two-players zero sum. H =heads and T =tails.

r r r r

��

H T H T

There are no optimal strategies.

Intuition of mixed strategies.

1.6.- Examples

Matching PenniesTwo-players zero sum. H =heads and T =tails.

r r r r

��

H T H T

1.6.- Examples

r r r r

��

H T H T

1.6.- Examples

r r r r

��

H T H T

1.6.- Examples

Strategic Voting

Three players f1, 2, 3g and three alternatives fx , y ,?g.Preferences: xP1?P2y ?P2xP2y yP3xP3?.Rules. Sequential Majority voting :

First, they will decide between x and y .

If x wins then they will decide between x and ?.If y wins then they will decide between y and ?.

First round: x . Second round: x .

Player 2 may anticipate this and vote for y in the �rst round. Then, yis selected in the �rst round and ? is �nally chosen (and ?P2x).Player 3 may anticipate this and vote for x in the �rst round. Then, xis selected in the �rst round and x is �nally chosen. He is better o¤since xP3?. The outcome is again x (the same than under sincerevoting). But now 2 and 3 are �lying�. However, this is an equilibrium(given the others�behavior, no player wants to change his behavior).

1.6.- Examples

Strategic VotingThree players f1, 2, 3g and three alternatives fx , y ,?g.

Preferences: xP1?P2y ?P2xP2y yP3xP3?.Rules. Sequential Majority voting :

First round: x .

Second round: x .

1.6.- Examples

Strategic VotingThree players f1, 2, 3g and three alternatives fx , y ,?g.Preferences: xP1?P2y ?P2xP2y yP3xP3?.

Rules. Sequential Majority voting :

First round: x .

Second round: x .

1.6.- Examples

Strategic VotingThree players f1, 2, 3g and three alternatives fx , y ,?g.Preferences: xP1?P2y ?P2xP2y yP3xP3?.Rules. Sequential Majority voting :

First round: x .

Second round: x .

1.6.- Examples

First round: x .

Second round: x .

1.6.- Examples

First, they will decide between x and y .If x wins then they will decide between x and ?.

If y wins then they will decide between y and ?.

First round: x .

Second round: x .

1.6.- Examples

First, they will decide between x and y .If x wins then they will decide between x and ?.If y wins then they will decide between y and ?.

First round: x .

Second round: x .

1.6.- Examples

First round: x .

Second round: x .

1.6.- Examples

First round: x .

Second round: x .

1.6.- Examples

Player 2 may anticipate this and vote for y in the �rst round. Then, yis selected in the �rst round and ? is �nally chosen (and ?P2x).

Player 3 may anticipate this and vote for x in the �rst round. Then, xis selected in the �rst round and x is �nally chosen. He is better o¤since xP3?. The outcome is again x (the same than under sincerevoting). But now 2 and 3 are �lying�. However, this is an equilibrium(given the others�behavior, no player wants to change his behavior).

1.6.- Examples

Cournot equilibrium (1838)

Ru¢ n, R.J. �Cournot Oligopoly and Competitive Behavior,�Reviewof Economic Studies 38, 1971.

Consider a market with n �rms producing an homogeneous good atconstant unit cost of c Euros.

Let yi be the units produced and sold by �rm i = 1, ..., n, and let

Y =n∑i=1yi be the total quantity in the market.

Suppose that the aggregate inverse demand function p = D(Y ) is

p = maxfa� b � Y , 0g,

where p is the price of the good, a > c and b > 0.

1.6.- Examples

Cournot equilibrium (1838)Competitive Solution:

Firm i thinks that i is small, and behaves competitively by taking theprice as given:

Given p, choose yi in order to max p � yi � c � yi .The unique price at which the above problem has an interestingsolution is p = c .

But then, the individual production of each �rm is indeterminate (butsmall to justify the belief that p0(yi ) = 0).

Firms have zero pro�ts.

Hence, Y � = a�cb .

1.6.- Examples

Given p, choose yi in order to max p � yi � c � yi .

The unique price at which the above problem has an interestingsolution is p = c .

1.6.- Examples

Cournot equilibrium (1838)Cournot Equilibrium:

Maximization problem of �rm i : Giveny�i = (y1, ..., yi�1, yi+1, ..., yn) choose yi in order to

max(a� b �n∑j=1yj ) � yi � c � yi .

First order condition:

a� 2 � b � yi � b �∑j 6=i yj � c = 0. (1)

Second order condition: �2 � b < 0 (the pro�t function is strictlyconcave on yi ).

Look for a symmetric equilibrium: yj = y for all j = 1, ..., n.

1.6.- Examples

a� 2 � b � yi � b �∑j 6=i yj � c = 0. (1)

1.6.- Examples

a� 2 � b � yi � b �∑j 6=i yj � c = 0. (1)

1.6.- Examples

a� 2 � b � yi � b �∑j 6=i yj � c = 0. (1)

1.6.- Examples

a� 2 � b � yi � b �∑j 6=i yj � c = 0. (1)

1.6.- Examples

Condition (1) can be written as, a� 2by � b(n� 1)y � c = 0. Thus,the optimal production, the price, and the pro�ts, depending on thenumber of �rms n, are

y(n) =a� c

b � (n+ 1) p(n) =a+ n � cn+ 1

π(n) =(a� c)2b � (n+ 1)2 .

Note that limn!∞ y(n) = 0, limn!∞ Y (n) = a�cb , limn!∞ p(n) = c ,

and limn!∞ π(n) = 0.

The sequence of Cournot equilibria converges to the competitiveequilibrium.

This gives a non-cooperative (strategic) foundation to thecompetitive equilibrium.

1.6.- Examples

y(n) =a� c

b � (n+ 1) p(n) =a+ n � cn+ 1

π(n) =(a� c)2b � (n+ 1)2 .

1.6.- Examples

y(n) =a� c

b � (n+ 1) p(n) =a+ n � cn+ 1

π(n) =(a� c)2b � (n+ 1)2 .

1.6.- Examples

y(n) =a� c

b � (n+ 1) p(n) =a+ n � cn+ 1

π(n) =(a� c)2b � (n+ 1)2 .

1.6.- Examples

Gloves (Shapley)

Shapley, L.S. �The Solutions of a Symmetric Market Game,�Annalsof Mathematical Studies 40, 1959.

Three players f1, 2, 3g, 1 and 2 have a left glove while 3 has a rightglove.

A correct pair of gloves is worth 1 unit, zero otherwise.

Characteristic function v (the amount that each coalition canguarantee by itself):

v(f1g) = 0 v(f2g) = 0 v(f3g) = 0v(f1, 2g) = 0 v(f1, 3g) = 1 v(f2, 3g) = 1v(f1, 2, 3g) = 1.

Which one is a stable distribution of this unit of utility among thethree players?

Core (based on a blocking idea): C (v) = f(0, 0, 1)g.

1.6.- Examples

Gloves (Shapley)

1.6.- Examples

Gloves (Shapley)

1.6.- Examples

Gloves (Shapley)

1.6.- Examples

Gloves (Shapley)

1.6.- Examples

Gloves (Shapley)

1.6.- Examples

Gloves (Shapley)

Core (based on a blocking idea): C (v) = f(0, 0, 1)g.Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Introduction 66 / 73

1.6.- Examples

Gloves (Shapley)

Suppose x + y + z = 1, x , y , z � 0 and z < 1.

Then either x > 0 or y > 0.

Suppose y > 0. Since x + z < 1 = v(f1, 3g), 1 and 3 together cansplit 1 as follows: (x + y

2 , z +y2 ).

So, the coalition f1, 3g blocks (x , y , z).

The monopoly power of player 3 together with the competitionbetween 1 and 2 drives the outcome (0, 0, 1).

Is it however �fair� this distribution of the unit of utility? After all,player 3 alone gets zero, he needs a left glove.

1.6.- Examples

Gloves (Shapley)

Suppose x + y + z = 1, x , y , z � 0 and z < 1.Then either x > 0 or y > 0.

2 , z +y2 ).

1.6.- Examples

Gloves (Shapley)

2 , z +y2 ).

1.6.- Examples

Gloves (Shapley)

2 , z +y2 ).

1.6.- Examples

Gloves (Shapley)

2 , z +y2 ).

1.6.- Examples

Gloves (Shapley)

2 , z +y2 ).

1.6.- Examples

Gloves (Shapley)

Shapley, L.S. �Some Topics in Two-person Games,� in Advances inGame Theory, editors: M. Dresher, J. Shapley, and A. Tucker.Princeton: Princeton University Press, 1964.

Suppose that players arrive in a order and all orders are equally likely(= 1/6). Marginal contributions of each player for each order:

Order 1 2 3

123 0 0 1132 0 0 1213 0 0 1231 0 0 1312 1 0 0321 0 1 0

Sh(v) = 1/6 1/6 4/6

Properties? (Sh(v) /2 C (v)). Axiomatic characterization.

1.6.- Examples

Gloves (Shapley)

Order 1 2 3

123 0 0 1132 0 0 1213 0 0 1231 0 0 1312 1 0 0321 0 1 0

Sh(v) = 1/6 1/6 4/6

Properties? (Sh(v) /2 C (v)). Axiomatic characterization.

1.6.- Examples

Gloves (Shapley)

Order 1 2 3

123 0 0 1132 0 0 1213 0 0 1231 0 0 1312 1 0 0321 0 1 0

Sh(v) = 1/6 1/6 4/6

Properties? (Sh(v) /2 C (v)). Axiomatic characterization.Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Introduction 68 / 73

1.6.- Examples

Bankruptcy (Talmut, 2.000 years B.C. and Aumann and Maschler)

Aumann, R. and M. Maschler. �Game Theoretic Analysis of aBankruptcy Problem from the Talmud,� Journal of Economic Theory36, 1985.

A man dies, leaving debts d1, ..., dn adding up more than his state E .

How the state E should be divided among creditors?

Most modern laws says proportionally, i.e., xi =E �di

d1+...+dnfor each

i = 1, ..., n.

Aumann and Mascher (1985) re-interpret an example in the Talmut.

1.6.- Examples

d1+...+dnfor each

i = 1, ..., n.

1.6.- Examples

d1+...+dnfor each

i = 1, ..., n.

1.6.- Examples

d1+...+dnfor each

i = 1, ..., n.

1.6.- Examples

d1+...+dnfor each

i = 1, ..., n.

1.6.- Examples

d1+...+dnfor each

i = 1, ..., n.

1.6.- Examples

d1 = 100 d2 = 200 d3 = 300E 1 = 100 100/3 100/3 100/3 Equal divisionE 2 = 200 50 75 75 ?E 3 = 300 50 100 150 Proportional

Is there a unique rational to justify the three proposals?

Yes!, a consistent idea of the following solution of the problem(E = 1; d1 = 1, d2 = 1/2):

1 concedes 0 to 2, 2 concedes 1/2 to 1, plus equal division (1/4) ofthe remainder (1/2), hence,x1 = 1/2+ 1/4 = 3/4 and x2 = 0+ 1/4 = 1/4.Solution: (3/4, 1/4). It coincides with the Nucleolus (Schemeidler,1969) of the TU�game v de�ned as follows: for each S 2 2N ,

v(S) = maxfE �∑j /2S dj , 0g.

1.6.- Examples

1 concedes 0 to 2, 2 concedes 1/2 to 1, plus equal division (1/4) ofthe remainder (1/2), hence,

x1 = 1/2+ 1/4 = 3/4 and x2 = 0+ 1/4 = 1/4.Solution: (3/4, 1/4). It coincides with the Nucleolus (Schemeidler,1969) of the TU�game v de�ned as follows: for each S 2 2N ,

1.6.- Examples

1 concedes 0 to 2, 2 concedes 1/2 to 1, plus equal division (1/4) ofthe remainder (1/2), hence,x1 = 1/2+ 1/4 = 3/4 and x2 = 0+ 1/4 = 1/4.

Solution: (3/4, 1/4). It coincides with the Nucleolus (Schemeidler,1969) of the TU�game v de�ned as follows: for each S 2 2N ,

1.6.- Examples

f1, 2g xk3 , k = 1, 2, 3d1 = 100 d2 = 200

e1 = 200/3 100/3 100/3 100/3e2 = 125 50 75 75e3 = 150 50 100 150

1.6.- Examples

f1, 3g xk2 , k = 1, 2, 3d1 = 100 d3 = 300

e1 = 200/3 100/3 100/3 100/3e2 = 125 50 75 75e3 = 200 50 150 100

1.6.- Examples

f2, 3g xk1 , k = 1, 2, 3d2 = 200 d3 = 300

e1 = 200/3 100/3 100/3 100/3e2 = 150 75 75 50e3 = 250 100 150 50

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