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04/18/23 Coalition Formation Roadmap: Chattrakul Sombattheera

1

Game Theory

By

Chattrakul Sombattheera

Supervisors

A/Prof Peter Hyland & Prof Aditya Ghose


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Game theory

Analysis of problems of conflict and cooperation among independent decision-makers. Players, having partial control over outcomes of the game, are eager to finish the game with an outcome that gives them maximal payoffs possibleEmile Borel, a French mathematician, published several papers on the theory of games in 1921Von Neumann & Morgenstern’s The Game Theory and Economics Behavior in 1944A convenient way in which to model the strategic interaction problems eg. Economics, Politics, Biology, etc.


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The Games

Game = <Rules, Components>Rules: descriptions for playing gameComponents: A set of rational players A set of all strategies of all players A set of the payoff (utility) functions for each

combination of players’ strategies A set of outcomes of the game A set of Information elements


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Modeling Game

The rules give details how the game is played e.g. How many players, What they can do, and What they will achieve, etc.

Modeler study the game to find equilibrium, a steady state of the game where players select their best possible strategies.To find equilibrium = to find solution = to solve games


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Player and Rationality

Player can be a person, a team, an organization

In its mildest form, rationality implies that every player is motivated by maximizing his own payoff.

In a stricter sense, it implies that every player always maximizes his payoff, thus being able to perfectly calculate the probabilistic result of every strategy.


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Movement of the Game

Simultaneous: All players make decisions (or select a strategy) without knowledge of the strategies that are being chosen by other players.

Sequential: All players make decisions (or select a strategy) following a certain predefined order, and in which at least some players can observe the moves of players who preceded them

Games can be played repeatedly


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Information

Information is what the players know while playing games: All possible outcomes The payoff/utility over outcomes Strategies or actions used

An item of information in a game is common knowledge if all of the players know it and all of the players know that all other players know it


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Information

Complete information: if the payoffs of each player are common knowledge among all the playersIncomplete information: if the payoffs of each player, or certain parameters to it, remain private information of each player.Perfect Information: Each player knows every strategy of the players that moved before him at every point. Imperfect Information: If a player does not know exactly what strategies other players took up to a point.


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Strategies

Player I, SI = {x1, x2}

Player II, SII = {y1, y2, y3}

Player III, SIII = {z1, z2}

S is a set of 12 combinations of strategies

Each combination of strategy is an action (strategy) profile e.g. (x1, y2, z1)

I II III S

x1 y1 z1 x1,y1,z1

x1 y1 z2 x1,y1,z2

x1 y2 z1 x1,y2,z1

x1 y2 z2 x1,y2,z2

x1 y3 z1 x1,y3,z1

x1 y3 z2 x1,y3,z2

x2 y1 z1 x2,y1,z1

x2 y1 z2 x2,y1,z2

x2 y2 z1 x2,y2,z1

x2 y2 z2 x2,y2,z2

x2 y3 z1 x2,y3,z1

x2 y3 z2 x2,y3,z2


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Outcome, Utility

In general, outcome is a set of interesting elements that the modeler picks from the value of actions, payoffs, and other variables after the game terminates. Outcomes are often represented by action (strategy) profilesUtility represents the motivations of players. A utility function for a given player assigns a number for every possible outcome of the game with the property that a higher number implies that the outcome is more preferred. Utility functions may either ordinal in which case only the relative rankings are important, but no quantity is actually being measured, or cardinal, which are important for games involving mixed strategies


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Payoff

Payoffs are numbers which represent the motivations of players. Payoffs may represent profit, quantity, "utility," or other continuous measures (cardinal payoffs), or may simply rank the desirability of outcomes (ordinal payoffs).In most of this presentation, we assume that utility function assigns payoffs


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Variety of Game

Game can be modelled with variety of its components

We introduce Non-cooperative form game

Normal (strategic) form game Extensive form game

Cooperative form game Characteristic function game


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Normal (Strategic) Form Game

An n-person game in normal (strategic) form is characterised byA set of players N = {1, 2, 3, …, n}A set S = S1 x S2 x … x Sn is the set of

combinations of strategy profiles of n playersUtility function ui : S R of each player


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Normal (Strategic) Form GameComponents of a normal form game can be represented in game matrix or payoff matrix

Game matrix of 2 players: Player I and Player II Each player has a finite number of strategies

S1 = {s11, s12} S2={s21, s22}

Player II

Player I s21 s22

s11 u1(s11, s21), u2(s11, s21) u1(s11, s22), u2(s11, s22)

s12 u1(s12, s21), u2(s12, s21) u1(s12, s22), u2(s12, s22)


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Zero Sum Game

Von Neumann and Morgenstern studied two-person games which result in zero sum: one player wins what the other player loses

The payoff of player II is the negative value of the payoff of player I

=

Player II

Player I s21 s22

s11 (ua,-ua) (-ub, ub)

s12 (-uc, uc) (ud, -ud)

Player II

Player I s21 s22

s11 ua -ub

s12 -uc ud


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Matching Pennies

Player I & Player II: Choose H or T (not knowing each other’s choice) If coins are alike, Player II wins $1 from Player I If coins are different, Player I wins $1 from Player II

Player II

H T

Player I

H (-1,1) (1,-1)

T (1,-1) (-1,1)

Player II

H T

Player I

H -1 1

T 1 -1

=


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Pure Strategy

A prescription of decision for each possible situation is known as pure strategy

A pure strategy can be as simple as : Play Head, Play Tail

A pure strategy can be more complicated as : Play Head after wining a game

We refer to each of strategies of a player as a pure strategy


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Maximax Strategy

“Maximax principle counsels the player to choose the strategy that yields the best of the best possible outcomes.”Two players simultaneously put either a blue or a red card on the tableIf player I puts a red card down on the table, whichever card player II puts down, no one wins anythingIf player I puts a blue card on the table and player II puts a red card, then player II wins $1,000 from player IFinally, if player I puts a blue card on the table and player II puts a blue card down, then player I wins $1,000 from player II


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Maximax Strategy

With maximax principle, player I will always play the blue card, since it has the maximum possible payoff of 1,000. Player II is rational, he will never play the blue card, since the red card gives him 1,000 payoff. In such a case, if player I plays by the maximax rule, he will always lose.The maximax principle is inherently irrational, as it does not take into account the

other player's possible choices.

Maximax is often adopted by naive decision-makers such as young children.

Player II

Blue Red

Player I

Blue 1000 -1000

Red 0 0


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Battle of the Pacific

In 1943, the Allied forces received reports that a Japanese convoy would be heading by sea to reinforce their troops. The convoy could take on of two routes -- the Northern or the Southern route. The Allies had to decide where to disperse their reconnaissance aircraft -- in the north or the south -- in order to spot the convoy

as early as possible. The payoff matrix shows

payoffs expressed in the number of days of bombing the Allies could achieve

Japanese

North South

Allies

North 2 2

South 1 3


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Minimax Strategy

Minimax strategy is to minimize the maximum possible loss which can result from any outcome. To cause maximum loss to the Japanese, the Allies would like to go SouthTo avoid maximum loss, in case the Allies go South, the Japanese would go NorthIf the Japanese go North,

the Allies would go North to maximize their payoff

Japanese

North South

Allies

North 2 2

South 1 3


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Domination in Pure Strategy

Player I selects a row while Player II selects a column in response to each other for their maximum payoffs

Player II’s F strategy is always better than G no matter what strategy Player I selects

Strategy G is dominated by F, or F is a dominant

strategy

rational player never plays

dominated strategies.

Player II

E F G H

Player I

A 12 -1 1 0

B 5 1 7 -20

C 3 2 4 3

D -16 0 2 16


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Solving Pure Strategy

Player I selects a row while Player II selects a column in response to each other for their maximum payoffs

Player I selects D for maximum payoff (16), Player II selects E for his maximum payoff (-16)

Player I then selects A,

while Player II selects F

Player I selects C, while

Player II cannot improve

Player II

E F G H

Player I

A 12 -1 1 0

B 5 1 7 -20

C 3 2 4 3

D -16 0 2 16


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Pure Strategy: Saddle PointStrategies (C,F) is an equilibrium outcome, players have no incentives to leaveAt (C,F), I knows that he can win at least 2 while II knows that he can lose at most 2The value 2 at (C,F) is the minimum of its row and is the maximum of its columns—

it is call the Saddle point or the value of the game

The saddle point is the game’s equilibrium outcome

A game may have a number of saddle points of the same value

Player II

E F G H

Player I

A 12 -1 1 0

B 5 1 7 -20

C 3 2 4 3

D -16 0 2 16


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Mixed-Strategies: Odd or Even

A player can randomly take multiple actions (or strategies) based on probability— mixed strategies

Player I and Player II simultaneously call out one of the numbers one or two.

Player I wins if the sum

of the number is odd

Player II win if the sum

of the number is even

Note: Payoffs in dollars.

Player II

Player I one two

one -2 3

two 3 -4


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Solving Odd or Even

Suppose Player I calls ‘one’ 3/5ths of the times and ‘two’ 2/5ths of the times at random If II calls ‘one’, I loses 2 dollars 3/5ths of the times

and wins 3 dollars 2/5ths of the times. On average, I wins

-2(3/5) + 3(2/5) = 0 If II calls ‘two’, I wins 3 dollars 3/5ths of the times and

loses 4 dollars 2/5ths of the

times, On average, I wins

-3(3/5) – 4(2/5) = 1/5

Player2

Player1 one two

one (p=3/5) -2 3

two (p=2/5) 3 -4


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Solving Odd or Even

I win 0.20 on average every time II calls ‘two’

Can I fix this so that he wins no matter what II plays?

Player II

Player I one two

one -2 3

two 3 -4


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Equalizing Strategy

Let p be a probability Player I calls ‘one’ such that I wins the same amount on average no matter what II callsSince I’s average winnings when II calls ‘one’ and ‘two’ are -2p+3(1-p) and 3p-4(1-p), respectively. So… -2p + 3(1-p) = 3p-4(1-p) 3 – 5p = 7p – 4 12p = 7 p = 7/12

Player2

Player1 one two

one (p) -2 3

two (1-p) 3 -4


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Equalizing Strategy

Therefore, I should call ‘one’ with probability 7/12 and two with 5/12

On average, I wins

-2(7/12) + 3(5/12) = 1/12

or 0.0833 every play regardless of what II does.

Such strategy that produces the same average winnings no matter what the opponent does is called an equalizing strategy


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Minimax Strategy

In Odd or Even, Player I cannot do better than 0.0833 if Player II plays properlyFollowing the same procedure, II calls ‘one’ with probability 7/12 ‘two’ with probability 5/12

If I calls ‘one’, II’s average loss is -2(7/12) + 3(5/12) = 1/12If I calls ‘two’, II’s average loss is 3(7/12) – 4(5/12) = 1/121/12 is called the value of the game or the saddle pointMixed strategies used to ensure this are called optimal strategy or minimax strategy


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Minimax Theorem

A two person zero sum game is finite if both strategy set Si and Sj are finite sets.

For every finite two-person zero-sum game There is a number V, call the value of the game There is a mixed strategy for Player I such that I’s

average gain is at least V no matter what II does, and There is a mixed strategy for Player II such that II’s

average loss is at most V no matter what I does


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Non-Zero Sum Game

The sum of the utility is not zero

Prisoner Dilemma

Nash equilibrium

Chicken

Stag Hunt


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Prisoner Dilemma

Two suspects in a crime are held in separate cells

There is enough evidence to convict each one of them for a minor offence, not for a major crime

One of them has to be a witness against the other (finks) for convicting major crime If both stay quiet, each will be jailed for 1 year If one and only one finks , he will be freed while the

other will be jailed for 4 years If both fink, they will be jailed for 3 years


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Prisoner Dilemma

Utility function assigned u1(F,Q) = 4, u1(Q,Q) = 3, u1(F,F) = 1, u1(Q,F) = 0 u2(Q,F) = 4, u2(Q,Q) = 3, u2(F,F) = 1, u2(F,Q) = 0

What should be the outcome of the game? Both would prefer Q But they have incentive for being freed, choose F Prisoner II

Prisoner I Quiet Fink

Quiet (3, 3) (0, 4)

Fink (4, 0) (1, 1)


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Prisoner Dilemma

Prisoner I: Acting Fink against Prisoner II’s Quiet yields better payoff than Quiet. Fink is called the best strategy against Prisoner II’s Quiet

Prisoner I: Acting Fink against Prisoner II’s Fink yields better payoff than

Quiet. Fink is the best

strategy against Prisoner II’s

Fink

Prisoner II


Quiet (3, 3) (0, 4)

Fink (4, 0) (1, 1)


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Dominant Strategy

A dominant strategy is the one that is the best against every other player’s strategy.

Prisoner I: Fink is the dominant strategy

Prisoner II: Fink is the dominant strategy

Outcome (1,1) is called

dominant strategy

equilibrium

Prisoner2

Prisoner1 Quiet Fink

Quiet (3, 3) (0, 4)

Fink (4, 0) (1, 1)


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Nash Equilibrium

John Nash, the economics Nobel Winner.

An action (strategy) profile a = (a1, a2, a3, …, an) is combination of action ai, selected from player i strategy Si

Nash equilibrium is “an action profile a* with the property that no player i can do better by choosing an action different from ai

*, given that every other

player j adheres to aj*.”


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Nash Equilibrium & Strategies

Nash equilibrium is “an action profile a* with the property that no player i can do better by choosing an action different from ai

*, given that every other player j adheres to aj

*.”

Players = {I, II, III}

SI={x1, x2}, SII={y1, y2, y3}, SIII={z1, z2}

(x1, y2, z1) is a Nash Equilibrium if uI(x1, y2, z1) ≥ uI (x2, y2, z1) and uII(x1, y2, z1) ≥ uII (x1, y1, z1) and uII(x1, y2, z1) ≥ uII (x1, y3, z1) and uIII(x1, y2, z1) ≥ uIII (x1, y2, z2)

I II III S

x1 y1 z1 x1,y1,z1

x1 y1 z2 x1,y1,z2

x1 y2 z1 x1,y2,z1

x1 y2 z2 x1,y2,z2

x1 y3 z1 x1,y3,z1

x1 y3 z2 x1,y3,z2

x2 y1 z1 x2,y1,z1

x2 y1 z2 x2,y1,z2

x2 y2 z1 x2,y2,z1

x2 y2 z2 x2,y2,z2

x2 y3 z1 x2,y3,z1

x2 y3 z2 x2,y3,z2


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Nash Equilibrium

What is the equilibrium in Prisoner Dilemma?Usually, dominant equilibrium is Nash equilibriumBut, Nash Equilibrium

may not be dominant equilibrium

Prisoner II


Quiet (3, 3) (0, 4)

Fink (4, 0) (1, 1)


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Stag Hunt Game

Each of a group of hunters has two options: he may remain attentive to the pursuit of a stag, or catch a hareIf all hunters pursue the stag, they catch it and share it equallyIf any hunter devotes his energy to catching a hare, the stag escape, and the hare belongs to the defecting hunter aloneEach hunter prefers a share of the stag to a hare


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Stag Hunt & Equilibrium

A group of 2 hunters value payoffs are u1(stag, stag) = u2(stag, stag) = 2,

u1(stage,hare) = 0, u2(stage,hare) = 1,

u1(hare,stag) = 1, u2(hare,stag) = 0 and

u1(hare,hare) = u2(hare,hare) = 1

There are 2 equilibria

(stag, stag) and

(hare, hare)

Hunter II

Hunter I Stag Hare

Stag (2, 2) (0, 1)

Hare (1, 0) (1, 1)


43

Chicken

There are two hot ‘Gong teenagers, Smith and BrownSmith drives a V8 Commodore heading South down the middle of Princes Hwy, and Brown drives V8 Falcon up NorthWhen approaching each other, each has the choice to stay in the middle or swerveThe one who swerves is called “chicken” and loses face, the other claims brave-hearted prideIf both do not swerve, they are killedBut if they swerve, they are embarrassingly called chicken


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Chicken & Nash Equilibrium

The cardinal payoffs are u(stay, stay) = (-3,-3), u(stay, swerve) = (2,0), u(swerve, stay) = (0,2) and u(swerve, swerve) = (1,1)There is no dominant strategy but there are two pure strategy Nash equilibria (swerve, stay) and (stay, swerve)How do the players

know which equilibrium will be played out?

Brown

Smith Stay Swerve

Stay (-3, -3) (2, 0)

Swerve (0, 2) (1, 1)


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Chicken

In mixed strategies, both must be indifferent between swerve and stayLet p be the probability for Brown to stay-3p = 2p + 1(1-p) p = 1/4 = 0.25

The chance for being survival is 1 – (p * p)

1 – 0.0625 = 0.9375

Brown

Smith Stay (p) Swerve (1-p)

Stay (-3, -3) (2, 0)

Swerve (0, 2) (1, 1)


46

Game with No Equilibrium

Matching Pennies: Player 1 & Player 2 choose H or T (not knowing each other’s choice) If coins are alike, Player 2 wins $1 from Player 1 If coins are different, Player 1 wins $1 from Player 2

There is no Nash equilibrium pure strategy

There, however, is a Nash equilibrium mixed strategy where each player

plays head with probability 0.5

The average payoffs for both

players are 0

Player II

H T

Player I

H (-1,1) (1,-1)

T (1,-1) (-1,1)


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Nash Equilibrium

In equilibrium, each player is playing the strategy that is a "best response" to the strategies of the other players. No one has an incentive to change his strategy given the strategy choices of the others

Game may not have equilibrium

Game may have equilibria

Equilibrium is not the best possible outcome !!!


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Pareto Optimum

Named after Vilfredo Pareto, Pareto optimality is a measure of efficiencyAn outcome of a game is Pareto optimal if there is no other outcome that makes every player at least as well off and at least one player better offA Pareto Optimal outcome cannot be improved upon without hurting at least one player. Often, a Nash Equilibrium is not Pareto Optimal implying that the players' payoffs can all be increased.


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Equilibrium and Optimum

In Prisoner Dilemma, both players have incentives to leave {Fink, Fink}

One will earn more

but the other will

be worst off.

{Q, Q} is Pareto optimal

Nash equilibrium does not

guarantee optimality

Prisoner2

Prisoner1 Quiet Fink

Quiet (3, 3) (0, 4)

Fink (4, 0) (1, 1)


50

Equilibrium & Optimum

In Stag Hunt, there are 2 equilibria (stag, stag) and (hare, hare)

Only one of the equilibria is optimal

Hunter2

Hunter1 Stag Hare

Stag (2, 2) (0, 1)

Hare (1, 0) (1, 1)


51

Equilibrium & Optimum

In Chicken game, equilibria are (Swerve, Stay) and (Stay, Swerve)

Both of equilibria have one swerve and one stay

Both equilibria are Pareto optimalBrown

Smith Stay Swerve

Stay (-3, -3) (2, 0)

Swerve (0, 2) (1, 1)


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Extensive Form Game

Game in extensive form can be represented by a topological tree or game treeA topological tree is a collection of finite nodes Each node is connected by a link There is a unique sequence of nodes and links between

any pair of nodes Node C follows B if the sequence of links joining A to

C passes through B Node X is called terminal if no nodes follows X


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Extensive Form Game

An n-person game in extensive form is characterised byA tree T, with a node A called the starting point

of TA utility function, assigning an n-vector to each

terminal node of TA partition of the non-terminal nodes of T into

n + 1 sets, Si, called the player sets


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Extensive Form Game

A probability distribution, defined at each node of Si among the intermediate followers of this vertex.

For each player i, there is a sub-partition of Si into subsets Si

j called information set

For each information set Sij,

All nodes have the same number of outgoing linksEvery path from root to terminal nodes can cross Si

j

only once


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Extensive Form Game


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Parlor GameParlor game is an extensive form gameThe rule specify a series of well-defined movesA move is a point of decision for a given player from among a set alternatives.A particular alternative

chosen by a player at a given decision point is a choice.

Sequence of choices until the game is terminated is a play.


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A Modified Spade Game

For simplicity, a set of cards are reduced to aces, 2s, and 3s.A deck of cards is divided into suits, one of which (say clubs) is discarded.A second suit (spades) is shuffled and placed face down on the tableEach of the two players has in his hand a complete suitThe cards are valued: ace = 1, 2 = 2 and 3 = 3The spades are turned over one by one and each is bided by one of the players, the one capturing the larger value of spades wins (46.)The first spade is turned over then the player simultaneously bid for the spade with a card in his hand: the higher value winsIf a draw occurs, the winner of the next round takes the spades


58

Modified Spade Game


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Matching Pennies in Extensive form Game

Player 1 & Player 2: Choose H or T (not knowing each other’s choice)

If coins are alike, Player 2 wins $1 from Player 1

If coins are different, Player 1 wins $1 from Player 2

H T

H

H T

T


60

X-O in Extensive Form Game


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Cooperative Game

Players can communicate (negotiate)

Players can make binding agreement (forming coalition)

Players can make side payment (transferable utility)


62

Coalitions in Cooperative Game

N is a set of playersA coalition S is a subset of N, a set of all coalitions is denoted by SThe set N is also a coalition, specially called grand coalitionA coalition structure is a set CS = {S1, S2, …, Sm} which is a partition of N such that Sj , j = 1, 2, …, m Si Sj = for all i j S1 S2 … Sm = N


63

Coalitions in Cooperative Game

N = {1, 2, 3} is a set of players

All possible coalitions are S ={, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}}Coalition structures are {{1}, {2}, {3}}, {{1}, {2,3}}, {{1,2}, {3}}, {{1,3}, {2}} and {{1,2,3}}


64

Payoff in Cooperative Game

A game eventually terminates in an end-state i.e. outcome or coalition structure.The quantitative representation of an outcome to a player is a payoff xi.A collection of payoffs to all players is a payoff vector x = (x1, x2, x3, …, xn)A payoff configuration is a pair of a payoff vector and a coalition structure denoted by (x; CS) = {x1, x2, x3,…, xn; S1, S2, …, Sm}


65

Cooperative Game in Characteristic Function FormAn n-person game in characteristic function form is characterized by a pair (N:) whereN = {1, 2, …, n} is a set of players; n ≥ 2v : S → R is a characteristic function defining a

real value to each coalition S of N.

Thus the game is named Characteristic Function Game (CFG)


66

Characteristic Function Game

Implicit assumptions:The value of any coalition is in money, and the players prefer more money to lessA coalition S forms by making a binding agreement on the way its value v(S) is to be distributed among its members. The amount v(S) does not in anyway depend on the actions of N-S, though N-S might partition it self into coalitions. The amount v(S) cannot given to any player outside S.


67


The characteristic function v is known to all players. Any agreement concerning the formation and disbursement of value is known to all n players as soon as it is made.

The characteristic function influences player affinities for each other.

Every nonempty coalition can form.


68


Odd Man Out, an example of CFG:Three players {1, 2, 3} bargain in pairs to form

a deal, dividing money, depending on coalitions If 1 and 2 combine, excluding 3, they split $4.0 If 1 and 3 combine, excluding 2, they split $5.0 If 2 and 3 combine, excluding 1, they split $6.0


69


Odd Man Out’s characteristic function:v({1}) = v({2}) = v({3}) = v({1,2,3}) = 0v({1,2})=4, v ({1,3})=5, v ({2,3})=6

Possible payoff configurations (2.0, 2.0, 0: {1,2},{3}) (2.5, 0, 2.5: {1, 3}, {2}) (0, 3.0, 3.0: {1}, {2,3}) (0, 0, 0: {1}, {2}, {3})


70


Sandal makers:Maker 1 and 2 make only left sandals, each at

rate 17 pieces at a timeMaker 3, 4 and 5 make only right sandals, each

at rate 10 pieces at a timeAny single sandal worth nothing while a pair

(of left and right!) sells $20.A coalition is a binding agreement between left

and right sandal makers.


71


Sandal makers characteristic function: v({1}) = v({2}) = v({3}) = v({4}) = v({5}) = 0 v({1,2}) = v({3,4}) = v({3,5}) = v({4,5}) = v({3,4,5}) = 0 v({1,3}) = v({2,3}) = v({1,4}) = v({2,4}) = v({1,5}) = v({2,5}) =

200 v({1,2,3}) = v({1,2,4}) = v({1,2,5}) = 200 v({1,3,4}) = v({1,3,5}) = v({1,4,5}) = v({2,3,4}) = v({2,3,5}) =

v({2,4,5}) = 340 v({1,3,4,5}) = v({2,3,4,5}) = 340 v({1,2,3,4}) = v({1,2,3,5}) = v({1,2,4,5}) = 400 v({1,2,3,4,5}) = 600


72

Characteristic Function Game (CFG)

Possible payoff configurations:(100, 100, 100, 100, 0: {1,3}, {2,4}, {5})(100, 100, 100, 100, 0: {1,4}, {2,3}, {5})(113.3, 100, 113.3, 113.3, 100: {1,3,4}, {2,5})(100, 100, 100, 100, 0:{1,2,3,4},{5})(120, 120, 120, 120, 120: {1,2,3,4,5})


73

Super-Additive Environment

A game is super-additive if v(S T) v(S) + v(T) for all S, T N such that S T = Ø.In a super-additive environment, e.g. sandal makers or social welfare, players tend to form a grand coalition.In a non-super-additive environment, self-interested players make the game very interesting.


74

Stability and Efficiency in CFG

The game is played and is meant to reach a stable state: no player has incentive to leave coalition or change strategyCore.

The issue of how well/fair the payoffs are distributed is efficiencyShapley value .


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Imputation in Super-additive

Von Neumann & Morgenstern believed the distribution of coalition values is the key to coalition formation.Let the value of a singleton coalition of player i v(i) is denoted by vi, payoff vectors should hold Individual rationality: xi vi for all i Collective rationality: xi = v(N)

An imputation is a payoff vector x = (x1, x2, … xn) satisfying xi vi and xi = v(N) for all i


76

Imputation in Sandal Makers

(100, 100, 100, 100, 0: {1,3}, {2,4}, {5})(100, 100, 100, 100, 0: {1,4}, {2,3}, {5})(113.3, 100, 113.3, 113.3, 100: {1,3,4}, {2,5})(100, 100, 100, 100, 0:{1,2,3,4},{5})(120, 120, 120, 120, 120: {1,2,3,4,5})


77

Modified Sandal Makers

All maker make 15 sandals, characteristic functions are v({1}) = v({2}) = v({3}) = v({4}) = v({5}) = 0 v({1,2}) = v({3,4}) = v({3,5}) = v({4,5}) = v({3,4,5}) = 0 v({1,3}) = v({2,3}) = v({1,4}) = v({2,4}) = v({1,5}) = v({2,5}) =

300 v({1,2,3}) = v({1,2,4}) = v({1,2,5}) = 300 v({1,3,4}) = v({1,3,5}) = v({1,4,5}) = v({2,3,4}) = v({2,3,5}) =

v({2,4,5}) = 300 v({1,3,4,5}) = v({2,3,4,5}) = 300 v({1,2,3,4}) = v({1,2,3,5}) = v({1,2,4,5}) = 600 v({1,2,3,4,5}) = 600


78

Imputation of Modified Sandal Makers

Payoff configuration (150, 150, 150, 150, 0: {1,2,3,4}, {5}) (150, 150, 150, 0, 150: {1,2,3,5}, {4}) (150, 150, 0, 150, 150: {1,2,4,5}, {3}) (120, 120, 120, 120, 120: {1,2,3,4,5})


79

Imputation, Domination and the Core

Imputation x dominates y over S N if xi > yi for all i in S and xi v(S)

The core is the set of all undominated imputations in the gameOnly imputations in the core can persist in pre-game negotiationsThe core can be empty


80

The Core of Sandal Makers

(100, 100, 100, 100, 0: {1,3}, {2,4}, {5})(100, 100, 100, 100, 0: {1,4}, {2,3}, {5})(113.3, 100, 113.3, 113.3, 100: {1,3,4},

{2,5})(100, 100, 100, 100, 0:{1,2,3,4},{5})(120, 120, 120, 120, 120: {1,2,3,4,5})


81

Core of Modified Sandal Makers

Payoff configuration (150, 150, 150, 150, 0: {1,2,3,4}, {5}) (150, 150, 150, 0, 150: {1,2,3,5}, {4}) (150, 150, 0, 150, 150: {1,2,4,5}, {3}) (120, 120, 120, 120, 120: {1,2,3,4,5})


82

Shapley Value

In a game (N,v), Shapley proposed a concept of fair distribution of payoff = (1, 2, ..., n), which is captured in three axioms. I: should only depend on v, if players i and j have

symmetric roles then i = j

II: If v(S) = v(S - i) for all coalition S N, then i = 0. Adding a dummy player i does not change the value j for other players j in the game

III: If (N, v) and (N, w) are two different games, and the sum game v + w is defined as (v + w)(S) = v (S) + w (S) for all coalitions S, then [v+w] = [v] + [w]


83

Shapley Value

To calculate the payoff, consider the players forming grand coalition step by step

Start by one player and add each additional player

As each player joins, award the new player an additional value he contributes to the coalition

Once this is done for each of the n! grand coalitions divide the accumulated awards to each player by n! to give the fair imputation


84

Shapley Value

Consider the following game. v(A) = v(B) = v(C) = 0 v(AB) = 2, v(AC) = 4, v(BC) = 6 v(ABC) = 7

The 6 (3!) ordered grand coalitions are: Order

Value added byA B C

ABC 0 2 5ACB 0 3 4BAC 2 0 5BCA 1 0 6CAB 4 3 0CBA 1 6 0

8 14 20


85

Shapley Value

Considering the ordered BCA coalition, the value added by each player is:B: v(B) - v() = 0 - 0 = 0C: v(BC) - v(B) = 6 - 0 = 6A: v(BCA) - v(BC) = 7 - 6 = 1

v(A) = v(B) = v(C) = 0 v(AB) = 2, v(AC) = 4, v(BC) = 6 v(ABC) = 7

= 1/6(8, 14, 20) = (1.33, 2.33, 3.33)

Order Value added by

A B C

ABC 0 2 5

ACB 0 3 4

BAC 2 0 5

BCA 1 0 6

CAB 4 3 0

CBA 1 6 0

8 14 20


86

Done !

Questions – Comments…?

2/06/2015 coalition formation roadmap: chattrakul sombattheera1 game theory by chattrakul...

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