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Solution concepts on cooperative games

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Cooperation vs. noncooperation. Two different models deal with the same problem ???? ... Minus: never be entirely sure that one supplies the right simple-minded answer ... – PowerPoint PPT presentation

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Title: Solution concepts on cooperative games


1
Solution concepts on cooperative games
  • Chih Chang
  • National Tsing Hua University

2
Overview
  • Cooperative games vs. non-cooperative games
  • Why is justification of a solution necessary?
  • Definition
  • Axiomatic approach
  • Nash program

3
Example 1
  • What will be the outcome of the following game?

4
Example 2
  • 3-person constant-sum game

5
Example 3
  • 1 seller and 2 buyers
  • 1 has a horse worthless to him
  • The value of the horse is 90 to 2
  • The value of the horse is 100 to 3

6
Cooperative games
  • a TU game
  • the set of players
  • a coalition
  • and

7
  • The imputation set
  • Each element in X(v) is called an imputation

8
non-cooperative game
  • A game in strategic form
  • , the player set.
  • , is the strategy
    space of player i
  • , is the utility
    function of player i.

9
  • Define the characteristic function
  • such that
  • and
  • a cooperative game

10
  • Cooperation vs. noncooperation
  • Two different models deal with the same problem
    ????
  • Play cooperatively? Play non-cooperatively?
  • Preplay or pre-negotiation and reach agreement
  • whether players sign the contract to make their
    agreement binding

11
  • Ideas to design solution concepts
  • Noncooperation equilibrium concept
  • Cooperation Justice, equity, fairness, stability

12
  • We call is a Nash
    equilibrium if

13
Solutions of cooperative games
  • The Shapley value
  • The idea of the Shapley value
  • expected value of marginal contribution
  • Why do we accept it?
  • Justification is necessary

14
  • Given a game ,
  • The excess
  • measures "dissatisfaction" of coalition S with
    respect to the payoff vector
  • order the excesses of in nonincreasing
    order

15
  • The prenucleolus
  • Minimize total dissatisfaction of payoff vector
    in the sense of lexicographic order
  • Involves an obscure notion interpersonal
    comparison of utility
  • Justification is necessary

16
The EANSC value
  • The EANSC value

17
  • Three ways to justify whether a solution is sound
  • by definition
  • if it not sound, its function is to locate points
  • Axiomatization
  • The difficult part is to find convincing axiom
  • the Nash program
  • bargaining or non-cooperative viewpoint

18
How to do Axiomatization?
  • To justify a solution from axiomatic viewpoint,
    we need
  • (1) List convincing properties, or axioms
  • (2 ) the solution satisfies all these axioms
  • (3) the solution is the only one satisfying all
    these axioms
  • (4) All axioms are independent

19
Basic axioms
  • Denote G the set of cooperative games.
  • A solution s defined on G is a function such that
    for each
  • is single-valuedness (SIVA)
  • is Pareto optimal (PO)

20
  • satisfies equal treatment property (ETP)
    if
  • are called substitutes in v.
  • The solution has ETP if are
    substitutes and , then

21
  • Covariance(COV) Let
  • For convenience, we will call PO, SIVA, ETP, COV
    to be Basic axioms

22
  • If a solution satisfies the basic axioms, then it
    is the the standard solution on 2-person games.
  • For i1,2
  • As , almost all famous solutions are
    the standard solution

23
Idea of consistency
  • Given a solution s, , and N?N
  • Players in N\N received the payoff assigned by
  • and
  • N\N satisfied and left,
  • N also agree with what players in N\N get
  • Player in N do not happy with the payoff
  • N would like to reallocate the amount x(N)

24
  • How to reallocate?
  • By devising a game to do reallocation
  • The reduced game
  • to reevaluate the worth of every subcoalition of
    N which depend on
  • A solution satisfies consistency
  • never a need to reallocate the payoff it has
    chosen.

25
  • That is, reallocate x(N) by playing the
    reduced game, the solution is still
  • Consistency refelcts "internal stability" or
    robustness of a solution.
  • If we restrict 2, we call it bilateral
    consistency.

26
  • The Shapley value (Hart and Mas-Collel)
  • Basic axioms and self-consistency
  • The prenucleolus (Sobolev, Orshan)
  • Basic axioms and max-consistency
  • The EANSC value ( Moulin)
  • Basic axioms and complement consistency

27
What is Nash Program?
  • Historical background
  • Nash bargaining problem (S,d)
  • Nash Solution
  • Independent of Irrelevant Alternatives (IIA)
  • Symmetry
  • COV
  • PO

28
  • Nash cooperative and noncooperative theory not
    rivals.
  • Instead, provide complementary insights.
  • the plausibility of the axioms is open to
    question, for instance, IIA

29
  • Constructing noncooperative bargaining models to
    test
  • The idea a bargaining game whose rules are
    explicitly specified in detail.
  • various plans open to the players when they are
    negotiating as strategies

30
  • To check whether the equilibrium outcomes of the
    bargaining games is the Nash solution
  • This line of research is referred to as Nash
    program.

31
  • Aumann(New Palgrave)
  • "How will rational agents come to an agreement in
    a bargaining situation?
  • Given the bargaining regulations, and a precise
    notion of the players' rationality, players set
    their strategies to bargain.

32
  • Each profile of strategies determines a whole
    bargaining process.
  • Equilibrium state reflects rational moves of
    players to reach a consensus.
  • In words, the idea of non-cooperative models of
    cooperative games has come to be known as the
    Nash program.

33
Nash program summary
  • A non-cooperative game is proposed to reflect
  • an institutional arrangements and negotiation
    protocols that lead to the solution as a
    consequence of self-interested-noncooperative-rati
    onal play.
  • A non-cooperative interpretation of a solution

34
Guideline to design Nash program
  • Step 1, devise a bargaining rule fair to each one
  • represented in extensive game form G
  • Step 2, Each player proposes a strategy
  • Step 3, claim the strategy profile to be NE or
    SPE of G
  • Step 4, the outcome of NE or SPE is the desired
    solution

35
Game Form G
  • Stage 1 players propose simultaneously
  • player i proposes
  • p(1) becomes the current proposer,

36
  • If then is the
    binding proposal and the game goes to Stage 3
  • If ,
    the game goes to Period 1 of Stage 2

37
  • Stage 2 Assume the binding proposal is y and
    the current proposer is i. There are at most
  • periods to
    go.
  • At node the current proposer proposes an
    allocation.
  • At node every player responds to accept or
    to reject and an permutation

38
  • There are two possibilities
  • The proposal is accepted by all, the proposal
    becomes the current proposal and the game goes to
    Stage 3.
  • Suppose not.
  • If , the game returns to
    the node
  • If the game ends.

39
  • Player p(1) gets and others get

40
  • Stage 3 The current proposer has two actions to
    take.
  • If L is taken, y is the payoff of the game. The
    game ends.
  • If C is taken, he has to choose a player, say j,
    in N\i., the game ends with the outcome

41
The purpose of the device
  • To avoid the first mover advantage
  • The current proposer is determined randomly by
    submitting a permutation simultaneously
  • Every player can be the current proposer by
    proposing a proper permutation
  • How many periods to go at Stage 2 is also part of
    strategy of players by proposing an integer at
    the beginning of the game

42
  • To reflect bilateral consistency
  • the current proposer has the privilege to adjust
    his payoff by taking L or C is the key
  • Note that everyone can be the current proposer
  • by deviation and submitting a specific
    permutation given others

.
43
Converse consistency
  • Converse consistency is an equilibrium payoff
    if every pair of players is in equilibrium
  • The device also satisfies converse consistency

44
  • Theorem For every , there is a
    NE s of G such that the outcome of s is
  • Theorem Every NE outcome of G is in

45
Comments
  • Binmore
  • cooperative game theory
  • plus offers simple-minded answers to
    simple-minded questions. Hence, easy for
    applications
  • Minus never be entirely sure that one supplies
    the right simple-minded answer to the right
    simple-minded question.

46
  • noncooperative game theory
  • Plus it cuts no corners and so, if an analysis
    leads to an unambiguous conclusion, one can be
    confident that the problem is genuinely solved.
  • Minus the conclusion only applies to one
    specific game.

47
  • Aumann
  • Nash program
  • "detail distracts attention from essentials
  • traditional cooperative game approach
  • by abstracting away from details yields
    valuable perspective".
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