introduction to game theory: cooperative games (2)cslui/csc6480/cooperative_game2.pdf · question:...

Introduction to Game Theory:Cooperative Games (2)

John C.S. Lui

Department of Computer Science & EngineeringThe Chinese University of Hong Kong

www.cse.cuhk.edu.hk/∼cslui

John C.S. Lui (CUHK) Advanced Topics in Network Analysis 1 / 39

Outline

Outline

1 Strategic Equivalence

2 Two Solution ConceptsStable sets of imputationsShapley Values


Strategic Equivalence

IntroductionConsider two games ν and µ in characteristic function form.Suppose the number of players is the same for both games. Thequestion: when can we say that ν and µ are essentially the same ?For example, if we simply change the unit of payoff from one gameto the other, the game is the same and we are simply multiplyingthe characteristic function by a positive constant.Another modification is that each player Pi received a fixedamount ci independently on how he plays.Since the player can do nothing to change the ci ’s, they would playas if these fixed amounts where not present.



Combining the above modifications, we have:

DefinitionLet ν and µ be the two games in characteristic function form with thesame number N of players. Then µ is strategically equivalent to ν ifthere exist constants k > 0, and c1, . . . , cN , such that, for everycoalition S

µ(S) = kν(S) +∑Pi∈S

ci . (1)

Note that ν and µ play symmetric roles, we can also express

ν(S) = (1/k)µ(S) +∑Pi∈S

(−ci/k).



ExampleThe game whose normal form appear in Table 1 has characteristicfunction:

ν(P) = 1, ν(∅) = 0,ν({P1,P2}) = 1, ν({P1,P3}) = 4/3, ν({P2,P3}) = 3/4,

ν({P1}) = 1/4, ν({P2}) = −1/3, ν({P3}) = 0.

Let k = 2 and c1, c2, and c3 be -1, 0, -1 respectively, we have µ,which is strategically equivalent to ν with

µ(P) = (2)1 + (−1 + 0− 1) = 0, µ(∅) = (2)0 = 0,µ({P1,P2}) = (2)1 + (−1 + 0) = 1,

µ({P1,P3}) = (2)(4/3) + (−1− 1) = 2/3, µ({P2,P3}) = 1/2,µ({P1}) = −1/2, µ({P2}) = −2/3, µ({P3}) = −1.

In this example, µ is zero-sum.John C.S. Lui (CUHK) Advanced Topics in Network Analysis 6 / 39


TheoremIf ν and µ are strategically equivalent, and ν is inessential, then so is µ.Thus if ν is essential, so is µ.

ProofAssuming if ν is inessential, compute

N∑i=1

ν({Pi}) =N∑

i=1

(kν({Pi}) + ci)

= kN∑

i=1

ν({Pi}) +N∑

i=1

ci = kν(P) +N∑

i=1

ci = µ(P).

This shows that µ is inessential. By symmetry, if µ is inessential, so isν. This if ν is essential, so is µ.



Relationship between "Equivalence" and "Imputations"

TheoremLet ν and µ be strategically equivalent N−person games. Then wehave

An N−tuple x is an imputation for ν if and only if kx + c is animputation for µ.An imputation x dominates an imputation y through a coalition Swith respect to ν if and only if kx + c dominates ky + c withrespect to µ through the same coalition.An N−tuple x is in the core of ν if and only if kx + c is in the coreof µ.



ProofSuppose that x is an imputation for ν. Then, for 1 ≤ i ≤ N,

µ({Pi}) = kν({Pi}) + ci ≤ kxi + ci ,

which verifies individual rationality since kxi + ci is the i th

component of kx + c.For collective rationality, we have

µ(P) = kν(P) +N∑

i=1

ci = kN∑

i=1

xi +N∑

i=1

ci =N∑

i=1

(kxi + ci) .

Therefore, kx + c is an imputation for µ.The converse of this statement is true because of the symmetry ofν and µ.The other two statements of the theorem are proved in a similarway.



RemarkThe previous theorem tells us that if we are studying a game incharacteristic function form, then we are simultaneously studyingall games which are strategically equivalent to it.In case ν and µ are strategically equivalent, then we use thephrase “ν and µ are the same up to strategic equivalence”.Another implication is that we can replace a game by another onewhose characteristic function is particularly easy to work with.



DefinitionA characteristic function µ is in (0,1)−reduced form if both thefollowing hold:

µ({P}) = 0 for every player P.µ(P) = 1.

ObservationA game in (0,1)−reduced form is obviously essential.Conversely, it is also true that, up to strategic equivalence, everyessential game is in (0,1)−reduced form.



TheoremIf ν is an essential game, then ν is strategically equivalent to a game µin (0,1)−reduced form.

ProofDefine

k =1

ν(P)−∑N

i=1 ν({Pi})≥ 0,

and for i ≤ i ≤ N, define

ci = −kν({Pi}).

Then µ is defined by Equation (1).It is easy to verify that µ is in (0,1)−reduced form.



ExampleLet us consider the game whose normal form is given in Table 1.

From the previous theorem, we have:

k =1

1− (−1/12)=

1213, and

c1 = −(12/13)(1/4) = −3/13, c2 = 4/13, c3 = 0.

Then µ is given by:

µ(P) = 1, µ(∅) = 0,µ(any one-player coalition) = 0,

µ({P1,P2}) = (12/13)(1)− 3/13 + 4/13 = 1,µ({P1,P3}) = µ({P2,P3}) = 1.



Example: continueFor this game µ, we immediately observe three things:

All three two-person coalitions are equally good.If a two-person coalition forms, the players will probably divide thepayoff equally (since the players have completely symmetricroles).There is no advantage to a two-player coalition in bringing in thethird party to form a grand coalition.Conclusion:

One of the two-player coalitions will form, the player will split thepayoff, the third player is left out in the cold.Prevailing imputations will be either (1/2,1/2,0), (1/2,0,1/2) or(0,1/2,1/2).Our analysis cannot predict which coalitions will actually form.



Example: continueIf we transform these conclusions back into terms of ν, then

one of the two-player coalitions will form.The prevailing imputation will be one of the three possibilitieswhich can be computed using the relationship between ν and µ(will be shown later)

Another exampleFor the 3-player game G, the (0,1)−reduced form µ is

µ({P1,P2}) = 3/8, µ({P1,P3}) = µ({P2,P3}) = 1/2.In this game, the nice symmetry is lost and we can safely say thatthe grand coalition is likely to form.To make a guess about what the final imputation might behazardous since we discussed that the core is large.



TheoremSuppose µ and ν are N−person games in (0,1)−reduced form. If theyare strategically equivalent, then they are all equal.

ProofBy definition of strategic equivalence, there exist constant k > 0, andc1, . . . ,CN such that

µ(S) = kν(S) +∑Pi∈S

ci

for every coalition S.

To prove that ν and µ are equal, we need to show that k = 1 andci = 0 for all i . Since both characteristic functions are zero for allone-player coalitions, we see that ci = 0, ∀i . Since both characteristicfunctions are 1 for the grand coalition, we see that k = 1.



Classification of Small Games

Up to strategic equivalence, the number of games with two or threeplayers is limited, as shown by the following three theorems.

TheoremA two-player game in characteristic function form is either inessentialor strategically equivalent to ν, where

ν(the grand coalition) = 1, ν(∅) = 0,ν(either one-player coalition) = 0.



Classification of Small Games: continue

In the case of constant-sum games with three players, we have

TheoremEvery three-player constant-sum game in characteristic function formis either inessential or strategically equivalent to ν, where

ν(the grand coalition) = 1, ν(∅) = 0,ν(any two-player coalition) = 1,ν(any one-player coalition) = 0.

Remark: It basically says that every essential constant-sum game withthree players is strategically equivalent to three-player, constant-sum,essential game in (0,1)-reduced form (or we called THREE). Forexample, the game given in Table 1 is strategically equivalent toTHREE.



Classification of Small Games: continue

Up to strategic equivalence, the three-player games which are notnecessarily constant-sum form a three-parameter family. We have

TheoremEvery three-player game in characteristic function form is eitherinessential or there exist constants a,b, c satisfying:

0 ≤ a ≤ 1, 0 ≤ b ≤ 1, 0 ≤ c ≤ 1,

such that the game is strategically equivalent to ν, where

ν(the grand coalition) = 1, ν(∅) = 0,

and

ν(any one-player coalition) = 0,ν({P1,P2}) = a, ν({P1,P3}) = b, ν({P2,P3}) = c.


Two Solution Concepts

Two approaches to find solution

As a solution concept for cooperative game, core has a problem sinceit is either a) empty, or b) there are so many imputations in the coreand we have no reasonable way to decide which ones are actuallylikely to occur.Two approaches are proposed, the are:

Stable sets of Imputations.Shapley Values


Two Solution Concepts Stable sets of imputations

DefinitionLet X be a set of imputations for a game in characteristic function form.Then we say that X is stable if the following two conditions hold:

(Internal Stability): No imputation in X dominates any otherimputation in X through any coalition.(External Stability): In y is any imputation outside X , then it isdominated through some coalition by some imputation inside X .

The idea of stable sets of imputations was introduced by von Neumannand Morgenstern in 1944, and they argued that a stable set is asolution of the game.



Comments on Stable SetsNote that an imputation inside a stable set may be dominated bysome imputation outside. Of course, that outside imputation is, inturn, dominated by some other imputations inside (via externalstability). So transitive property does not hold for imputationdominance.Since all imputations inside X are equal, one which actuallyprevails would be chosen in some way, say, via pure chance,custom,..etc. But there is a problem since there may be animputation outside X which dominates a given imputation inside.And if a coalition is formed based on this outside imputation, X isnot stable anymore.Someone argue that this chaotic series of formation anddissolutions of coalitions of different stable sets is ok, since reallife often looks that way.



ExampleFor the three-person game in Table 1, we have showed (via(0,1)-reduced form), that it has a stable set. Let

X = {(0,1/2,1/2), (1/2,0,1/2), (1/2,1/2,0)}.

We have the following result

TheoremThe set X defined above is a stable set for THREE.



ProofWe denote the characteristic function for THREE by µ.We first verify the internal stability.

By symmetry it is enough to show that the imputation (0,1/2,1/2)does not dominate (1/2,0,1/2) through any coalition.The only possible coalition through which this domination couldoccur is {P2}, but µ({P2}) = 0 < 1/2, and this violates thefeasibility condition in the definition of dominance.

To verify about external stability, let y be an outside imputation:We must show that one of the members of X dominates it throughsome coalition.Note that there are at least two values of i for which yi < 1/2. If thiswere not true, we would have yi ≥ 1/2 for two values of i .Since each yi is nonnegative, and

∑i yi = 1, this implies y is one of

the imputation in X . This contradicts the assumption that y 6∈ X .By symmetry, we may assume that y1 and y2 are both less than1/2, but then (1/2,1/2,0) dominates y through coalition {P1,P2}.



CommentNote that there are imputations outside X which dominatemembers of X .Consider, for example, (2/3,1/3,0) dominates (1/2,0,1/2) throughcoalition {P1,P2}.On the other hand, (0, 1/2, 1/2) (a member of X ) dominates (2/3,1/3, 0) through {P2,P3}.



Stable Set for any games ∈ THREESince every essential constant-sum game with three players isstrategically equivalent to THREE, one can use X to obtain thestable set for any such game.Let ν be the game whose normal form is shown in Table 1. Then

µ(S) = kν(CS) +∑Pi∈S

ci ,

for every coalition S, where k = 12/13, c1 = −3/13, c2 = 4/13and c3 = 0.Thus ν(S) = (1/k)µ(S) +

∑Pi∈S(−ci/k).

Replacing each imputation x in X by (1/k)x − (1/k)c, gives us astable set for ν, namely

{(19/24,5/24,0), (19/24,−1/3,13/24), (1/4,5/24,13/24)}.


Two Solution Concepts Shapley Values

Shapley ValuesProposed by L.S. Shapley in 1953, an interesting attempt todefine, in a fair way, an imputation which embodies what theplayers’ final payoffs "should" be.It takes into account a player’s contribution to the success of thecoalition she belongs to.If the characteristic function of the game is ν, and if S is thecoalition to which player Pi belongs, then

δ(Pi ,S) = ν(S)− ν(S − {Pi})

is a measure of the amount that Pi has contributed to S by joiningit.



Comment on δ(Pi ,S)

Note that δ(Pi ,S), by themselves, are not very revealing.Consider the game of THREE:

We have δ(Pi ,P) = 0 for any Pi : no one contributed anything.If S is any two-player coalition, then δ(Pi ,S) = 1 for each player inS. That is, the sum of contributions is greater than ν(S).

IntuitionNotice that once the players have collectively agreed on animputation, it might as well be assumed that it is the grandcoalition which forms.Why? Because the condition of collective rationality ensures thatthe total of all payments (via the imputation) is ν(P).We concentrate on the process by which the grand coalitioncomes into being: it starts with the first player, then the 2nd player,...etc. Or the process is characterized by an ordered list of players.



Example: four-person gameSuppose that its characteristic function is

ν(P) = 100, ν(∅) = 0,ν({P1}) = 0, ν({P2}) = −10, ν({P3}) = 10, ν({P4}) = 0,ν({P1,P2}) = 25, ν({P1,P3}) = 30, ν({P1,P4}) = 10,ν({P2,P3}) = 10, ν({P2,P4}) = 10, ν({P3,P4}) = 30,

ν({P1,P2,P3}) = 50, ν({P1,P2,P4}) = 30,ν({P1,P3,P4}) = 50, ν({P2,P3,P4}) = 40.

One ordering of the players through which the grand coalition couldform:

P3,P2,P1,P4.

The total number of ordering for the grand coalition to form: 4!. Ingeneral, if there are N players, we have N! possibilities. Each orderingoccurs with probability 1/N!.



Example: four-person game (continue)Given that the grand coalition forms according to the givenordering:

δ(P1, {P3,P2,P1}) = ν({P3,P2,P1})−ν({P3,P2}) = 50−10 = 40.

This is a measure of the contribution of P1 makes as she entersthe growing coalition.The Shapley value, φi , is this:

Make the same sort of calculation for each of the N! possibleorderings of the playersWeight each one by the probability of 1/N! of that ordering to occur.Add the results.

We will show how to derive the Shapley value φi :so that the computation of φi is somewhat easier.show that φ = (φ1, . . . , φN) is an imputation.



Shapley valueNote that out of the N! ordering, there are many duplications.Suppose Pi occurs at position k . Denote the S be the set of kplayers up to and including Pi in this ordering. If we permute thepart of the ordering coming before Pi , and permute the partcoming after Pi , we obtain a new ordering in which Pi again is inthe k th position. In any of these permuted ordering, we have

δ(Pi ,S) = ν(S)− ν(S − {Pi}).

There are (k − 1)! permutations of the players coming before Piand (N − k)! permutations of players coming after Pi , the termδ(Pi ,S) occurs (k − 1)!(N − k)! times.Finally, the Shapley value for Pi , or φi , is:

φi =∑Pi∈S

(N − |S|)!(|S| − 1)!

N!δ(Pi ,S). (2)



ExampleConsider the game whose normal form is given in Table 1 with thegiven characteristic function.To find φ1, there are four coalition containing P1:

{P1}, {P1,P2}, {P1,P3}, {P1,P2,P3}.

So Eq (2) has four terms in this case.We compute

δ(P1, {P1}) = 1/4− 0 = 1/4, δ(P1, {P1,P2}) = 1− (−1/3) = 4/3,δ(P1, {P1,P3}) = 4/3−0=4/3, δ(P1, {P1,P2,P3})=1−(3/4)=1/4.

Thenφ1 =

2!0!

3!

14

+1!1!

3!

43

+1!1!

3!

43

+0!2!

3!

14

=1118.

Similarly, φ2 = 136 , φ3 = 13

36 . So φ =(11

18 ,1

36 ,1336

)is an imputation.



Interpretation

Note that φ =(11

18 ,1

36 ,1336

)is an imputation.

P ′1s Shapley value is largest of the three, indicating that P1 is thestrongest.P ′2s Shapley value is very small.P3 is in the middle.A glance at the characteristic function supports this "value"



More examplesThe 3-player game G, we can compute the Shapley value and it is

(1/8,5/8,1/4).

This numbers seem to reasonably reflect the advantage thatplayer P2 has in the game.For the Used Car game, the Shapley values are:

φN = 433.333..., φA = 83.333, ... φM = 183.33....

Thus, Mitchel gets the car for $433.33, but has to pay Agnew$83.33 as a bride for not bidding against him. And the Shapleyvector indicates that Nixon is in the most powerful position.



TheoremLet ν be a game in characteristic function form. Then the Shapleyvector for ν is an imputations.

Proof:We prove “individual rationality”, we must show that φi ≥ ν({Pi}).By super-additivity, if Pi ∈ S,δ(Pi ,S) = ν(S)− ν(S − {Pi}) ≥ ν({Pi}). Thus,

φi ≥

∑Pi∈S

(N − |S|)!(|S| − 1)!

N!

ν({Pi}).

The sum in this inequality is the sum of the probabilities of thedifferent orderings of the players, and it must equal 1, soφi ≥ ν({Pi}).



Proof: continueTo prove “collective rationality”, consider

N∑i=1

φi =N∑

i=1

∑Pi∈S

(N − |S|)!(|S| − 1)!

N!δ(Pi ,S).

In the double sum, fix our attention on the terms involving ν(T ),where T is a fixed nonempty coalition which is not equal to P.There are two kinds of terms involving ν(T ), those with positivecoefficient (when T = S):

(N − |T |)!(|T | − 1)!

N!,

and those with a negative coefficient (when T = S − {Pi}):

−(N − 1− |T |)!|T |!N!

.



Proof: continueThe first occurs |T | times (one for each member of T ), and thesecond kind occurs N − |T | times (once for each player outsideT ).The coefficient of the double sum is:

|T |(N−|T |)!(|T |−1)!N! − (N−|T |)(N−1−|T |)!|T |!

N! =(N−|T |)!|T |!

N! − N−|T |)!|T |!N! = 0.

Therefore, the only term left in the double sum are those involvingthe grand coalition, and those involving the empty coalition. Sinceν(∅) = 0, we have

N∑i=1

φi =N(0!)(N − 1)!

N!ν(P) = ν(P).



Shapley value for The Lake Wobegon GameContinue


introduction to game theory: cooperative games (2)cslui/csc6480/cooperative_game2.pdf · question:...

Documents