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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 possible
- Emile Borel, a French mathematician, published several papers on the theory of games in 1921
- Von Neumann & Morgenstern’s The Game Theory and Economics Behavior in 1944
- A 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 game
- Components:
- 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 players
- Incomplete 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)

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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) profiles
- Utility 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 by

- A set of players N = {1, 2, 3, …, n}
- A set S = S1xS2x … xSn is the set of combinations of strategy profiles of n players
- Utility function ui: S R of each player

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Normal (Strategic) Form Game

- Components 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}

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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

=

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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

=

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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 table
- If player I puts a red card down on the table, whichever card player II puts down, no one wins anything
- If player I puts a blue card on the table and player II puts a red card, then player II wins $1,000 from player I
- Finally, 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.

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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

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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 South
- To avoid maximum loss, in case the Allies go South, the Japanese would go North
- If the Japanese go North,

the Allies would go

North to maximize their

payoff

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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.

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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

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Pure Strategy: Saddle Point

- Strategies (C,F) is an equilibrium outcome, players have no incentives to leave
- At (C,F), I knows that he can win at least 2 while II knows that he can lose at most 2
- The 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

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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.

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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

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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?

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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 calls
- Since 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

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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 properly
- Following 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) = 1/12
- If I calls ‘two’, II’s average loss is 3(7/12) – 4(5/12) = 1/12
- 1/12 is called the value of the game or the saddle point
- Mixed 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

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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

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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

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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)

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Nash Equilibrium

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

may not be dominant

equilibrium

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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 hare
- If all hunters pursue the stag, they catch it and share it equally
- If any hunter devotes his energy to catching a hare, the stag escape, and the hare belongs to the defecting hunter alone
- Each 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)

Coalition Formation Roadmap: Chattrakul Sombattheera

Chicken

- There are two hot ‘Gong teenagers, Smith and Brown
- Smith drives a V8 Commodore heading South down the middle of Princes Hwy, and Brown drives V8 Falcon up North
- When approaching each other, each has the choice to stay in the middle or swerve
- The one who swerves is called “chicken” and loses face, the other claims brave-hearted pride
- If both do not swerve, they are killed
- But 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?

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Chicken

- In mixed strategies, both must be indifferent between swerve and stay
- Let 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

Coalition Formation Roadmap: Chattrakul Sombattheera

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

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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 efficiency
- An 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 off
- A 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

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Equilibrium & Optimum

- In Stag Hunt, there are 2 equilibria (stag, stag) and (hare, hare)
- Only one of the equilibria is optimal

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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 optimal

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Extensive Form Game

- Game in extensive form can be represented by a topological tree or game tree
- A 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 by

- A tree T, with a node A called the starting point of T
- A utility function, assigning an n-vector to each terminal node of T
- A 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 Sij called information set
- For each information set Sij,
- All nodes have the same number of outgoing links
- Every path from root to terminal nodes can cross Sij only once

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Extensive Form Game

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Parlor Game

- Parlor game is an extensive form game
- The rule specify a series of well-defined moves
- A 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 table
- Each of the two players has in his hand a complete suit
- The cards are valued: ace = 1, 2 = 2 and 3 = 3
- The 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 wins
- If a draw occurs, the winner of the next round takes the spades

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Modified Spade Game

Coalition Formation Roadmap: Chattrakul Sombattheera

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

Coalition Formation Roadmap: Chattrakul Sombattheera

X-O in Extensive Form Game

Coalition Formation Roadmap: Chattrakul Sombattheera

Cooperative Game

- Players can communicate (negotiate)
- Players can make binding agreement (forming coalition)
- Players can make side payment (transferable utility)

Coalition Formation Roadmap: Chattrakul Sombattheera

Coalitions in Cooperative Game

- N is a set of players
- A coalition S is a subset of N, a set of all coalitions is denoted by S
- The set N is also a coalition, specially called grand coalition
- A coalition structure is a set CS = {S1, S2, …, Sm} which is a partition of N such that
- Sj, j = 1, 2, …, m
- SiSj = for all ij
- S1S2 … Sm = N

Coalition Formation Roadmap: Chattrakul Sombattheera

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}}

Coalition Formation Roadmap: Chattrakul Sombattheera

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}

Coalition Formation Roadmap: Chattrakul Sombattheera

Cooperative Game in Characteristic Function Form

- An n-person game in characteristic function form is characterized by a pair (N:) where
- N = {1, 2, …, n} is a set of players; n ≥ 2
- v : 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)

Coalition Formation Roadmap: Chattrakul Sombattheera

Characteristic Function Game

Implicit assumptions:

- The value of any coalition is in money, and the players prefer more money to less
- A 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.

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Characteristic Function Game

- 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.

Coalition Formation Roadmap: Chattrakul Sombattheera

Characteristic Function Game

- 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

Coalition Formation Roadmap: Chattrakul Sombattheera

Characteristic Function Game

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

Coalition Formation Roadmap: Chattrakul Sombattheera

Characteristic Function Game

- Sandal makers:
- Maker 1 and 2 make only left sandals, each at rate 17 pieces at a time
- Maker 3, 4 and 5 make only right sandals, each at rate 10 pieces at a time
- Any 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.

Coalition Formation Roadmap: Chattrakul Sombattheera

Characteristic Function Game

- 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

Coalition Formation Roadmap: Chattrakul Sombattheera

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})

Coalition Formation Roadmap: Chattrakul Sombattheera

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.

Coalition Formation Roadmap: Chattrakul Sombattheera

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 strategy
- Core.
- The issue of how well/fair the payoffs are distributed is efficiency
- Shapley value .

Coalition Formation Roadmap: Chattrakul Sombattheera

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 iv(i) is denoted by vi, payoff vectors should hold
- Individual rationality: xivi for all i
- Collective rationality: xi = v(N)
- An imputation is a payoff vector x = (x1, x2, … xn) satisfying xivi and xi = v(N) for all i

Coalition Formation Roadmap: Chattrakul Sombattheera

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})

Coalition Formation Roadmap: Chattrakul Sombattheera

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

Coalition Formation Roadmap: Chattrakul Sombattheera

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})

Coalition Formation Roadmap: Chattrakul Sombattheera

Imputation, Domination and the Core

- Imputation x dominates y over S N if
- xi > yi for all i in S and
- xiv(S)
- The core is the set of all undominated imputations in the game
- Only imputations in the core can persist in pre-game negotiations
- The core can be empty

Coalition Formation Roadmap: Chattrakul Sombattheera

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})

Coalition Formation Roadmap: Chattrakul Sombattheera

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})

Coalition Formation Roadmap: Chattrakul Sombattheera

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]

Coalition Formation Roadmap: Chattrakul Sombattheera

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

Coalition Formation Roadmap: Chattrakul Sombattheera

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

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:

Coalition Formation Roadmap: Chattrakul Sombattheera

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

Shapley Value- Considering the ordered BCA coalition, the value added by each player is:

B: v(B) - v() = 0 - 0 = 0

C: v(BC) - v(B) = 6 - 0 = 6

A: 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)

Coalition Formation Roadmap: Chattrakul Sombattheera

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