Game Theory. Lecture Notes By Y. Narahari. Department of Computer Science and Automation Indian Institute of Science Bangalore, India July 2012

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1 Game Theory Lecture Notes By Y. Narahari Department of Computer Science and Automation Indian Institute of Science Bangalore, India July COOPERATIVE GAME THEORY The Two Person Bargaining Problem Note: This is a only a draft version, so there could be flaws. If you find any errors, please do send to hari@csa.iisc.ernet.in. A more thorough version would be available soon in this space. Cooperation refers to coalitions of two or more players acting together with a common purpose. Since rationality and intelligence are two fundamental concepts in game theory, any cooperation has to be achieved in the presence of agents whose behavior is ultimately derived from the objective of maximizing their own individual payoffs. In fact, cooperative game theory has been developed without abandoning the individual decision theoretic foundations of game theory. This was the view point taken by John Nash [, ]. According to him, cooperation between players can be studied using the concepts of non-cooperative game theory since cooperative actions can be considered as the result of some process of bargaining among the cooperating players. In this bargaining process, each player should be expected to behave according to some bargaining strategy that satisfies the original utility-maximization criterion as in any other game situation. The essential idea of Nash is to define a cooperative transformation that transforms a game in extensive or strategic or Bayesian form into a game (which is also in extensive or strategic or Bayesian form) which represents the situation when the players have, in addition to the existing strategic options specified in the original game, certain other options for bargaining with the other players to jointly plan cooperative strategies. This is on the lines of what we have studied in the previous chapter on correlated strategies. Nash s Program We have already explained the motivation for cooperation through our discussion of correlated strategies, games with contracts, games with communication, correlated equilibria, etc. Let us recall the prisoner s dilemma problem, which is a classic example for illustrating the benefits of cooperation. NC C NC, 0, C, 0 5, 5

2 In the above situation, the rationality of the players suggests the equilibrium (C,C) which is not exactly an optimal option for the players. The two players here have a strong incentive to bargain with each other or sign a contract to transform this game into one which has equilibria that are better for both the players. Nash s program for cooperative game theory is to define cooperative solution concepts such that a cooperative solution for any given game is a Nash equilibrium of some cooperative transformation of the original non-cooperative game. The notion of Nash program was introduced by Nash in the classic paper: J.F. Nash. Two person cooperative games. Econometrica, Volume, pp. 8-40, 953. However, if we carefully describe all the possibilities that are feasible when the agents bargain or sign contracts with each other, we may end up with a game that has numerous equilibria. This means Nash s program may not lead to a unique cooperative solution. This entails a theory of cooperative equilibrium selection. According to Myerson, the foundations of cooperative game theory depend on the role of arbitration, negotiation, and welfare properties in determining the focal equilibrium in a game with multiple equilibria. The Two Person Bargaining Problem and the Nash Bargaining Solution We now study the classic two person bargaining problem enunciated by Nash in the following article. J.F. Nash. The bargaining problem. Econometrica, Volume 8, pp. 55-6, 950. According to Nash, the term bargaining refers to a situation in which individuals have the possibility of concluding a mutually beneficial agreement there is a conflict of interests about which agreement to conclude no agreement may be imposed on any player without that player s approval. The following assumptions are implicit in Nash s formulation: when two players negotiate or an impartial arbitrator arbitrates, the payoff allocations that the two players ultimately get should depend only on:. the payoffs they would expect if the negotiation or arbitration were to fail to reach a settlement, and. on the set of payoff allocations that are jointly feasible for the two players in the process of negotiation or arbitration. The two person bargaining problem has been applied in many important contexts:. Management Labor Arbitration where the management negotiates contracts with the labor union. International Relations, where two Nations or two groups of Nations work out agreements on issues such as nuclear disarmament, military cooperation, anti-terrorist strategy, etc.

3 3. Duopoly Market Games where two major competing companies work out adjustments on their production to maximize their profits 4. Supply Chain Contracts where a buyer and a supplier work out a mutually beneficial contract that stimulates a long-term relationship Description of the Problem The two person bargaining problem consists of a pair (F,v) where F is a closed, convex subset of R. F represents the set of feasible allocations and is called the feasible set. v = (v,v ) R. v represents the disagreement payoff allocation and is called the disagreement point. It is also called the status-quo point. This gives the payoffs for the two players in case the negotiations fail. It may be noted that v is invariably chosen to belong to the feasible set F. F {(x,x ) R : x v ;x v } is non-empty and bounded. Justification for the Assumptions. F is assumed to be convex. This can be justified as follows. Assume that the players can agree to jointly randomized strategies. Consequently, if the utility allocations x = (x,x ) and y = (y,y ) are feasible and 0 α, then the expected utility allocation αx + ( α)y can be achieved by planning to implement x with probability α and to implement y with probability ( α).. F is assumed to be closed (that is, all convergent sequences in F converge to a point in F). This is a natural topological requirement. 3. The set F {(x,x ) R : x v,x v } is assumed to be non-empty and bounded. This assumption asserts that there exists some feasible allocation that is at least as good as disagreement for both players, but unbounded gains over the disagreement point are not possible. Connection to Two Player Non-Cooperative Games Let Γ = ({,},S,S,u,u ) be a two person strategic form game. If the strategies of the players can be regulated by binding contracts, then one possibility for the set F is: F = {(u (α),u (α)) : α (S S )} where u i (α) = s S α(s)u i (s) If the players strategies cannot be regulated by binding contracts (that is, moral hazards are possible), then a possibility for F would be: F = {(u (α),u (α)) : α is a correlated equilibrium of Γ} The following are three possibilities for the disagreement point v. 3

4 . The first possibility is to let v i be the minimax value for player i. v = min σ (S ) max u (σ,σ ) σ (S ) v = min σ (S ) max u (σ,σ ) σ (S ). Let (σ,σ ) be some focal Nash equilibrium of G and let v = u (σ,σ );v = u (σ,σ ) 3. Derive v = (v,v ) from some rational threats (to be explained later). A sound theory of negotiation or arbitration would ideally allow us to identify, given any bargaining problem (F,v), some allocation vector in R which we denote by f(f,v) that would be selected as a result of negotiation or arbitration. Thus the problem is one of finding an appropriate solution function f from the set of all two-person bargaining problems into R, which is the set of payoff allocations. The Axioms of Nash Nash s approach to this problem was axiomatic. Nash first came up with a list of properties a reasonable bargaining solution function is expected to satisfy and then exhibited a unique solution that satisfies all of these properties. Nash axiomatized five different properties:. Strong Efficiency. Individual Rationality 3. Scale Covariance 4. Independence of Irrelevant Alternatives (IIA) 5. Symmetry Some Notation and Definitions Let f i (F,v) denote the payoff allocation to player i (i =,). That is, we have: f(f,v) = (f (F,v),f (F,v)). Let us use the following notation for vectors x = (x,x ) and y = (y,y ): x y x y and x y x > y x > y and x > y Given a feasible set F, we say x F is strongly Pareto efficient iff there exists no other point y F such that y x and y i > x i for at least one player i. x F is said to be weakly Pareto efficient iff there exists no point y F such that y > x. 4

5 Axiom : Strong Efficiency f(f,v) should belong to F and also, for any x F, x f(f,v) x = f(f,v) Alternatively, f(f,v) F and there does not exist any x F such that x f(f,v) with x i > f i (F,v) for at least one i {,}. This axiom The axiom asserts that the solution to any two person bargaining problem should be feasible and strongly Pareto efficient. This implies that there should be no other feasible allocation that is better than the solution for one player and not worse than the solution for the other player. Axiom : Individual Rationality This asserts that f(f,v) v. This implies that neither player should get less in the bargaining solution than he/she could get in disagreement. Axioms and are illustrated in Figure F (v, v) f(f,v) Figure : Strong efficiency and individual rationality Axiom 3: Scale Covariance For any numbers λ,λ,µ,µ with λ > 0, λ > 0, if G = {(λ x + µ,λ x + µ : (x,x ) F } 5

6 and then w = (λ v + µ,λ v + µ ), f(g,w) = (λ f (F,v) + µ,λ f (F,v) + µ ) In the above, the bargaining problem (G,w) can be derived from (F,v) by applying increasing affine utility transformations which will not affect any decision theoretic properties of the utility functions. The axiom implies that the solution of (G,w) can be derived from the solution of (F,v) by the same transformation. This axiom is illustrated in Figure. u 00 (λ f + µ, G (λ v λ + µ, v + µ ) f(f, v) 00 F (v, v) λ f + µ ) u Figure : Illustration of scale co-variance Axiom 4: Independence of Irrelevant Alternatives For any closed convex set G, G F and f(f,v) G f(g,v) = f(f,v) This axiom says that eliminating feasible alternatives (other than the disagreement point) that would not have been chosen should not affect the solution. This axiom is illustrated in Figure 3. If an arbitrator were to choose a solution by maximizing some aggregate measure of social gain, that is, f(f,v) = argmax x F M(x,v) where M(x,v) is a measure of social gain by choosing x instead of v, then Axiom 4 will always be satisfied. 6

7 u f(g, v) = f(f, v) G (v, v) F u Figure 3: Independence of irrelevant alternatives Axiom 5: Symmetry This axiom implies that if the positions of players and are completely symmetric in the bargaining problem, then the solution should also treat them symmetrically. This axiom is illustrated in Figure 4. The Nash Bargaining Solution v = v and {(x,x ) : (x,x ) F } = F f (F,v) = f (F,v) Using brilliant arguments, Nash showed the existence and uniqueness of the Nash bargaining solution which satisfies the five axioms. This is stated in the form of the following theorem. Theorem: There is a unique solution function f(.,.) that satisfies Axioms () through (5). The solution function satisfies, for every two person bargaining problem (F,v), f(f,v) argmax (x,x ) F x v ;x v (x v )(x v ) 7

8 u f(f, v) F (v, v) Symmetric F u Figure 4: Illustration of symmetry An Example to Illustrate Nash Axioms Figure shows a closed convex space F which is actually the convex hull enclosing the points (4,0), (,), and (0,4). Suppose the default point is (,). It can be shown that the Nash bargaining solution is (,). This illustrates Pareto efficiency, individual rationality, and symmetry. A Proof of the Nash Bargaining Theorem We shall define space G obtained through the following scaling, λ = λ = ;µ = µ =, consider the Nash bargaining problem (G,w) with w = (.5,.5) that is obtained as w = λ v + µ,λ v + µ ). Using scale invariance, Nash bargaining solution becomes (,). The problems (G, w) and (F, v) also illustrate the IIA axiom. We find that G F, G is closed convex, and (,) G. Therefore we have f(g,w) = f(f,v) = (,). Finally, the problem (H,(0.5,0.5)) illustrates scale co-variance with λ = λ = ;µ = µ = 0. A function f : S R where S is non-empty and convex is said to be quasi-concave if x,y S 8

9 x x = x (,) (0,0) (,3) G (3,) x H F Figure 5: An example to illustrate Nash axioms and λ [0,], f(λx + ( λ)y) min(f(x),f(y)) f is strictly quasi-concave if the above inequality is strict. A strictly quasi-concave function has a unique optimal (maximum) solution. The function (x v )(x v ) is strictly quasi-concave and it has a unique maximizer. Consider the optimization problem: max (x v ) (x v ) subject to (x,x ) F (convex and closed) x v x v where (v,v ) F. This optimization problem has a unique optimal solution. Call this solution as (x,x ). Let F R be the convex and closed. Suppose v = (v,v ) F and F {(x,x ) R : x v ;x v } is non-empty and bounded. Suppose the function f(f, v) satisfies the five axioms () strong efficiency () individual rationality (3) scale covariance (4) independence of irrelevant alternatives (5) symmetry. 9

10 Then the Nash bargaining theorem says that f(f,v) = (x,x ) ) which happens to be the unique solution of optimization problem above. Proof: First we show that the optimal solution of the (x,x ) satisfies all the five axioms (part ). Next we show that if the solution f(f,v) satisfies all the five axioms, then f(f,v) = (x,x ) (part ).. Proof of part : (.) Strong efficiency : We have to show that ( ˆx, ˆx ) F such that ˆx x and ˆx x with at least on inequality strict. Suppose such a ( ˆx, ˆx ) exists. Since the objective function N( ˆx, ˆx ) > N(x,x ) which is not possible since N(x,x ) ) is the maximum possible value of N(x,x ) in the optimization problem. (.) Individual rationality is immediately satisfied being one o the constraints in the optimization problem. (.3) Scale co-variance: For λ > 0,λ > 0,µ,µ, define Consider the problem G = {λ x,+µ,λ x,+µ ) : (x,x ) F } max (y,y ) G (y (λ v + µ ))(y (λ v + µ )) This can be written using y = λ x + µ and y = λ x + µ as max (x,x ) F (λ x + µ (λ v + µ ))(λ x + µ (λ v + µ )) The above problem is the same as which attains maximum at (x,x ). Therefore attains maximum at (λ x + µ, λ x + µ ) max (x,x ) F λ λ (x v )(x v ) max (y,y ) G (y (λ v + µ )) (y (λ v + µ )) (.4) Independence of Irrelevant : Alternatives, we are given G F with G closed and convex. Let (x,x ) be optimal to F and v and let (y,y ) be be optimal to G and v. It is also given that (x,x ) G. 0

11 Since (x,x ) is optimal to F which is a superset of G, we have N(x,x ) N(y,y ) Since (y,y ) is optimal to G and (x,x ) G, we have Therefore we have N(y,y ) N(x,x ) N(x,x ) = N(y,y ) Since the optimal solution is unique, we then immediately get (.5) Symmetry : Suppose we have (x,x ) = (y,y ) {(x,x ) : (x,x ) F } = F and v = v. Since v = v, then (x,x ) maximizes (x v )(x v ). Since the optimal solution is unique, we should have (x,x ) = (x,x ) which immediately yields x = x ).. Proof of Part We first prove for essential bargaining problems which are problems such that there exists at least one allocation y F such that y > v and y > v. We are given that f(f v) is a bargaining solution that satisfies all the five axioms. We have that x > v and x > v. Consider the following transformation where In other words, we have L(x,x ) = (λ x + µ,λ x + µ ) λ = x v ;λ = x v µ = v x v ;µ = v x v L(x,x ) = ( x v x v, x v x v ) Note that L(v,v ) = (0,0) and L(x,x ) = (,). Defining G = {L(x,x ) : (x,x ) F }, we have thus transformed the problem (F,(v,v )) to the problem (G,(0,0)). In fact, since L(x,x ) = (,), it is known that the transformed problem (G,(0,0)) has its solution at (,). See figure. Also observe that the objective function of transformed problem is (x 0)(x 0) = x x. First we show that x + x (x x ) G To show this, we assume that x + x + and arrive at a contradiction. Suppose α (0,) and t = ( α)(,) + α(x x ) = ( α + αx, α + αx ) Since G is convex and (,) G, we therefore have t G and we get t t = ( α + αx )( α + αx ) = ( α) + α x x + ( α)α(x + x )

12 If x + x >, we then get t t > ( α) + α x x + ( α)α = ( α) + α x x We can always choose α sufficiently small and make t t >, which is impossible since the optimal value of t t is. G is bounded and we can always find a rectangle H which is symmetric about the line x = x such that G H and H is also convex and bounded. Further choose H such that the point (,) G is on the boundary of H. Strong efficiency and symmetry imply that f(h,(0,0)) = (,) We can now use independence of irrelevant alternatives to get f(g,(0,0)) = (,) We know G is obtained through scaling. Now use scale co-variance to get f(g,(0,0)) = L(f(f,v)) we know that... L(f(f,v)) = (,) L(x,x ) = (,)... L(f(f,v)) = (x,x ) Alternative Proof of Nash s Result First we prove the theorem for a special class of two person bargaining problems called essential bargaining problems and subsequently, we generalize this to the entire class of problems. A two person bargaining problem (F,v) is said to be essential if there exists at least one allocation y F that is strictly better for both the players than the disagreement allocation v : i.e., y > v and y > v. Proof for Essential Bargaining Problems Let (F,v) be any essential two person bargaining problem, so there exists some y F such that y > v and y > v. We will show that to satisfy axioms () - (5), f must select the allocation that maximizes the Nash product. Let x F be a point in F that achieves the maximum of the Nash product (x v )(x v ) over all x in F such that x v. We will show that x is the same as f(f,v). Since the problem is essential, the Nash product (x v )(x v ) will surely attain positive values for at least some point in F. Since the point x achieves the maximum value of the Nash product, it is easy to see that x > v.

13 x x x = 000 (0,) (,) H (0,0) 00 G (v v ) x *, x* F (,0) x + x < x Figure 6: Diagram showing F, G, H for proving Nash theorem Maximizing (x v ) (x v ) is equivalent to maximizing log(x v )(x v ) which is a strictly concave function of x, therefore having a unique maximum. Thus the maximizing point x is unique given F and v. Let us scale the problem (F,(v,v )) to (G,(0,0). Let λ = x v ; λ = x v µ = v x v ; µ = v x v 3

14 Define a function L : R R such that That is, L(x,x ) = (λ x + µ,λ x + µ ) ( ) L(x,x ) = x v (x v ), x v (x v ) Note that L(v,v ) = (0,0) and therefore Nash product for this new transformed problem is (z 0)(z 0) = z z. See Figure 5. z 00 E F F 0 (v, v) 00 G 00 0 (0,0) G 000 E 000 slope = Hyperbola E z z = z z + z = Figure 7: Proof of Nash s results for essential bargaining problems Let G = {L(x) : x F }. This is the feasible set of the transformed problem. For any x R, z = L(x) z z = λ λ (x v )(x v ) 4

15 and λ λ is a positive constant. Note that ( ) ( ) x v x v z z = x v x v Thus, since x maximizes the Nash product over F, L(x ) must maximize the product z z over all z G. Now, L(x ) = (,) is the maximum value of z z over the set G. Consider the hyperbola {z R : z z = } The equation of the tangent to this hyperbola at (,) is: z = z or z + z =. Thus the hyperbola has slope at the point (,). Therefore the convex set G lies below the line {z : z + z = }. The line {z : z + z = } which has slope and goes through the point (,) is tangent to the convex set G at the point (,). Define the convex set E = {z R : z +z }. Then G E. Note that E is symmetric while the set G need not be symmetric. See Figure 5 for a visualization of the different convex sets. To satisfy Axiom (strong efficiency) and Axiom 5 (symmetry), we need f(e,(0,0)) = (,) This is because E is a convex set and closed. Strong efficiency implies that the solution point should be on the frontier of z + z (it cannot lie in the interior). Symmetry will then imply that z = z. We see that f(e,(0,0)) G and G E. Further G is closed and convex. Therefore, to satisfy Axiom 4, we require f(g,(0,0)) = (,) We know that G is obtained from F through scaling. If the solution f(f,v) has to satisfy scale co-variance (Axiom 3), then, This implies that L(f(F,v)) = f(g,(0,0)) = (,) f(f,v) = x since L(x ) maximizes the Nash product of the Transformed problem (G,(0,0) (note that x maximizes the original Nash product x v )(x v ). Axiom (individual rationality) is trivially satisfied because we are dealing with essential bargaining problems. What we have shown is that to satisfy the five axioms, f must select the allocation that maximizes the Nash product (x v ) (x v ) among all individually rational allocations in F. This proves the result for essential bargaining problems. 5

16 Proof for Inessential Bargaining Problems Consider the case when the problem (F,v) is inessential, that is, there does not exist any y F such that y > v. Since F is convex, the above implies that there exists at least one player i such that y F, y v y i = v i. Note that if we could find y,z F such that y v,z v,y > v,z > v, then y + z would be a point in F that was strictly better than v for both the players. Let x be the allocation in F that is best for the player other than i subject to the constraint x i v i. This implies that vector x is the unique point that is strongly Pareto efficient in F and individually rational relative to v. This would mean that, to satisfy Axioms and, we must have f(f,v) = x. It can be easily observed that x achieves the maximum value of (x v )(x v ), which is zero for all individually rational allocations in this inessential bargaining problem. This shows that the five axioms can be satisfied by one and only one solution function f which always selects the unique, strongly efficient allocation that maximizes the Nash product over all feasible individually rational allocations. We now only need to show that the solution function f satisfies all the five axioms. This is left as an exercise. Note on Essential Bargaining Problems For essential bargaining problems, the strong efficiency axiom can be replaced by the following weak efficiency axiom: Axiom. (Weak Efficiency). f(f,v) F and there does not exist any y F such that y > f(f,v). Also, it is trivial to see that for essential bargaining problems, the individual rationality assumption is not required. Thus in the case of essential two person bargaining problems, any solution that satisfies Axioms,3,4,5 must coincide with the Nash bargaining solution. Example : Both Players Risk Neutral Let v = (0,0) and F = {(y,y ) : 0 y 30;0 y 30 y } The bargaining problem (F, v) can be thought of a situation in which the players can share 30 million rupees in any way in which they agree or get zero if they cannot agree, with both players being risk neutral (have linear utility for money). The Nash bargaining solution f(f,v) is the vector y = (y,y ) such that y 0, y 0, and y maximizes the Nash product (y v )(y v ). This can be easily seen to be (5,5) which looks perfectly reasonable. 6

17 Example : One Player Risk Neutral and the Other Risk Averse Let v = (0,0) and F = {(y,y ) : 0 y 30;0 y 30 y } This problem is similar to the one above, except that player risk neutral (has linear utility for money) player is risk averse (with a utility scale that is proportional to the square root of the money he gains) Note that d [ ] y 30 y = 0 dy yields y = 0. The Nash bargaining solution is the allocation (0, 0) = (0,3.6). This corresponds to a wealth sharing of (0,0). Other Solutions: Egalitarian and Utilitarian Solutions In typical bargaining situations, interpersonal comparisons of utility are made in two different ways. Principle of Equal Gains: Here a person s argument will be: you should do this for me because I am doing more for you. This leads to an egalitarian solution. Principle of the Greatest Good: The argument here goes as follows: You should do this for me because it helps me more than it hurts you. This leads to a utilitarian solution. For a two-person bargaining problem (F,v), the egalitarian solution is the unique point (x,x ) F that is weakly efficient in F and satisfies the equal gains condition: x v = x v ((x,x ) F is said to be weakly efficient if there does not exist any (y,y ) F such that y > x and y > x ). A utilitarian solution is any solution function that selects, for every two person bargaining problem (F,v), an allocation (x,x ) F such that x + x = max (y + y ) (y,y ) F If agents in negotiation or arbitration are guided by the equal gains principle, the natural outcome is the egalitarian solution. If it is guided by the principle of greatest good, a natural outcome is a utilitarian solution. The egalitarian and the utilitarian solutions violate the axiom of scale co-variance. The intuition for this as follows. The scale - covariance axiom is based on an assumption that only the individual decision theoretic properties of the utility scales should matter. Also interpersonal comparisons of utility have no decision-theoretic significance as long as no player can be asked to decide between being himself or someone else. 7

18 λ-egalitarian Solution Consider a two person bargaining problem (F,v). Given numbers λ,λ,µ,µ, with λ > 0, λ > 0, let L(y) = (λ y + µ,λ y + µ ) for y R Given the problem (F,v), define L(F) = {L(y) : y F } Then the egalitarian solution of (L(F), L(v)) is L(x), where x is the unique weakly efficient point in F such that λ (x v ) = λ (x v ) This is called the λ-egalitarian solution of (F,v). If λ = (,), this is called the simple egalitarian solution. The egalitarian solution does not satisfy scale co-variance because the λ-egalitarian solution is generally different from the simple egalitarian solution. λ-utilitarian Solution A utilitarian solution of (L(f),L(v)) is a point L(z) where z = (z,z ) is a point in F such that λ z + λ z = max (λ y + λ y ) (y,y ) F The solution point z is called a λ-utilitarian solution of (F, v). Utilitarian solutions do not satisfy scale co-variance because a λ-utilitarian solution is generally not a simple utilitarian solution. Relationship to the Nash Bargaining Solution Note that the equal gains principle suggests a family of egalitarian solutions and the greatest good principle suggests a family of utilitarian solutions. These solutions correspond to application of these principles when the payoffs are compared in a λ-weighted utility scale. As λ increases and λ decreases, the λ-egalitarian solutions trace out the individually rational, weakly efficient frontier moving in the direction of decreasing payoff to player. Also as λ increases and λ decreases, the λ-utilitarian solutions trace out the entire weakly efficient frontier, moving in the direction of increasing payoff to player. It turns out that for an essential two person bargaining problem (F,v), there exists a vector λ = (λ,λ ) such that λ > (0,0) and the λ-egalitarian solution of (F,v) is also a λ-utilitarian solution of (F,v). λ and λ that satisfy this property are called natural scale factors. Remarkably, the allocation in F that is both λ-egalitarian and λ-utilitarian in terms of the natural scale factors is the Nash bargaining solution. Thus the Nash bargaining solution can be viewed as a natural synthesis of the equal gains and greatest good principles. The following theorem formalizes this fact. 8

19 Theorem Let (F, v) be an essential two person bargaining problem. Suppose x is an allocation vector such that x F and x v. Then x is the Nash-bargaining solution for (F,v) iff there exist strictly positive numbers λ and λ such that λ x λ v = λ x λ v and λ x λ v = max y F (λ y + λ y ) Proof: Let H(x,v) denote the hyperbola H(x,v) = {y R : (y v )(y v ) = (x v )(x v )} The allocation x is the Nash bargaining solution of (F,v) if and only if the hyperbola H(x,v) is tangential to F at x. Now, the slope of the hyperbola H(x,v) at x This means H(x,v) is tangent at x to the line for any two positive numbers λ and λ such that = (x v ) (x v ) {y R : λ y + λ y = λ x + λ x } λ (x v ) = λ (x v ) Therefore x F is the Nash bargaining solution of (F,v) if any only if F is tangent at x to a line of the form {y R : λ y + λ y = λ x + λ x } for some (λ,λ ) λ (x v ) = λ(x v ). Example 3 Let v = (0,0) and F = {(y,y ) : 0 y 30;0 y 30 y }. The above problem is the same as Example. Recall that the Nash bargaining solution is the allocation (0, 0) = (0,3.6) which corresponds to a monetary allocation of 0 for player and 0 to player. Note that the risk averse player is under some disadvantage as per the Nash bargaining solution. The natural scale factors for this problem are λ = λ = 40 =

20 Let player s utility for a monetary gain of D be 6.35 D instead of D. This is a decisiontheoretically irrelevant change. Let player s utility be measured in the same units as money gained (λ = ). The representation of this bargaining problem becomes (G,(0,0)) where G = {(y,y ) : 0 y 30;0 y y } Now the Nash bargaining solution is (0, 0), which still corresponds to a monetary allocation of 0 to player and 0 for player. Note that this is both an egalitarian solution and a utilitarian solution. Games with Transferable Utility Let Γ = N,(S i ),(u i ) be a strategic form game. Informally, Γ is said to be a game with transferable utility if, in addition to the strategy options listed in S i, each player i can. give any amount of money to any other player, or. simply destroy money Each unit of net monetary outflow decreases the utility of player i by one unit. This is formalized in the following definition. Definition: A strategic form game Γ = N,(S i ),(u i ) is called a game with transferable utility if it can be represented by the game G = N,(Ŝi),(û i ) where for i N, Ŝ i = S i R n + û i ((s j,t j ) j N ) = u i ((s j ) j N ) + [ j i(t j (i) t i (j))] t i (i) where t j = (t j (k)) k N. The monetary transfer t j (k), for any k j, represents the quantity of money given by player j to player k and t j (j) denotes the amount of money destroyed by j but not given to any other player. An Example Let N = {,}. The payoff function û i would look like the following: Note that û (s,s,t (),t (),t (),t ()) = u (s,s ) (t () t ()) t () û (s,s,t (),t (),t (),t ()) = u (s,s ) (t () t ()) t () û (s, s, t (), t (), t (), t ()) + û (s, s, t (), t (), t (), t ()) = u (s, s ) + u (s, s ) t () t () Note in the above definition that û i is linearly dependent on the transfers t j. Thus risk neutrality is implicitly assumed when we talk of transferable utility in a game. Transferable utility is an assumption which ensures that the given scale factors in a game will also be the natural scale factors for the Nash bargaining solution. 0

21 Let (F,v) be a two person bargaining problem derived from a game with transferable utility. Let v represent the maximum transferable wealth that the players can jointly achieve. Then the feasible set F will be of the form F = {(x,x ) R : x + x v } This implies that, in the presence of transferable utility, a two person bargaining problem can be fully characterized by three numbers.. v = disagreement payoff to player. v = disagreement payoff to player 3. v = total transferable wealth available to the players if they cooperate. x will be a Nash bargaining solution to this problem iff λ > 0, λ > 0 such that λ x λ v = λ x λ v λ x + λ x = max (λ x + λ x ) (x,x ) F To satisfy the above conditions, λ = λ, since otherwise max (λ x + λ x ) = + (x,x ) F since both x and x are real numbers and x + x v will allow unbounded values for x and x separately. Thus the conditions for f(f,v) become f (F,v) v = f (F,v) v f (F,v) + f (F,v) = v Solving these equations, the following general formulae can be obtained for the Nash bargaining solution of a game with transferable utility. f (F,v) = v + v v f (F,v) = v + v v Let us say F is derived under the assumption that the players strategies can be regulated by binding contracts. Then we can say, as a consequence of the Transferable utility property, v = max σ (S) (u (σ) + u (σ)) Rational Threats We have shown that a two player bargaining problem (F,v), in a transferable utility setting, has the Nash bargaining solution f (F,v) = v + v v f (F,v) = v + v v

22 Note that the payoff to player increases as the disagreement payoff to player decreases. That is, hurting player in the event of disagreement may actually help player if a cooperative agreement is reached. Thus, reaching a cooperating point depends on a disagreement point and this induces rational players to behave in an antagonistic way in trying to create a more favorable disagreement point. This phenomenon is called the chilling effect. This is described formally by Nash s theory of rational threats. Nash s Theory of Rational Threats Let Γ = ({,},S,S,u,u ) be a two player finite game in strategic form. Let F be the feasible set derived from Γ. F will take the form F = {(u (σ),u (σ)) : σ (S)} where u i (σ) = s S σ(s)u i (s) for i =, in the case of binding contracts without non-transferable utility. In the case of binding contracts with transferable utility, F will take the form where F = {(y,y ) R : y + y v } v = max σ (S) [u (σ) + u (σ)] Before entering into the negotiation or arbitration, each player is required to choose a threat τ i (S i ). If the players fail to reach a cooperative agreement, then each player is committed independently to carry out his threat. If (τ,τ ) is a pair of threats chosen by the two players, the disagreement point in the two person bargaining problem should be (u (τ,τ ),u (τ,τ )) Let w i (τ,τ ) be the payoff that player i (i =,) gets in the Nash bargaining solution with the above disagreement point. That is, w i (τ,τ ) = f i (F,(u (τ,τ ),u (τ,τ ))) The game Γ = ({,}, (S ), (S ),w,w ) is called the threat game corresponding to the original game Γ. Assume that the players expect that they will finally reach a cooperative agreement, that will depend on the disagreement point, according to the Nash bargaining solution. This implies that the players should not be concerned about carrying out their threats but instead should evaluate their threats only in terms of their impact on the final cooperative agreement. This makes each player i to choose his threat τi so as to maximize w i (τ,τ ) given the other player s expected threat. This motivates to call (τ,τ ) a pair of rational threats iff w (τ,τ ) w (σ,τ ) σ (S ) w (τ,τ ) w (τ,σ ) σ (S )

23 Thus (τ,τ ) is a Nash equilibrium of the threat game Γ = ({,}, (S ), (S ),w,w ) Kakutani s fixed point theorem can be used to prove the existence of rational threats. In the threat game, Player wants to choose his threat τ so as to put the disagreement point (u (τ,τ ),u (τ,τ )) as favorable to him as possible and as unfavorable to Player as possible. Player wants choose his threat τ so as to put the disagreement point as favorable to him as possible and as unfavorable to Player as possible. The payoffs in the threat game (in the transferable utility case) are where w (τ,τ ) = v + u (τ,τ ) u (τ,τ ) w (τ,τ ) = v + u (τ,τ ) u (τ,τ ) v = max σ (S) [u (σ) + u (σ)] Note that v is a constant. Therefore maximizing w (τ,τ ) is equivalent to maximizing u (τ,τ ) u (τ,τ ). Similarly maximizing w (τ,τ ) is equivalent to maximizing u (τ,τ ) u (τ,τ ). It is straightforward to note that, under the transferable utilities case, τ and τ are rational threats for players and iff (τ,τ ) is an equilibrium of the two person zero sum game Γ = ({,}, (S ), (S ),u u,u u ) The game Γ is called the difference game derived from G. Example to Illustrate Different Ways of Choosing Disagreement Point In this example from Myerson s book, we examine three different ways to determine a disagreement point for deriving the Nash bargaining solution.. Equilibrium of game Γ. Minimax values 3. Rational threats Consider the following two player strategic form game a b a 0, 0 5, b 0, 5 0, 0 Note that the maximum total payoff achievable in this game is v = 0. 3

24 Choice : Nash Equilibrium Note that (b,b ) is a non-cooperative equilibrium. Let us take the payoffs in the equilibrium as the disagreement point, that is, v = (0,0) The Nash bargaining solution is f(f,v) = (0,0) Choice : Minimax Values Minimax value for player : v = 0. This is achieved when player chooses an offensive threat b and player chooses a defensive response b. Minimax value for player : v =. This is achieved when player chooses a as his optimal offensive threat and player chooses b as an optimal defensive strategy. Therefore, v = (0, ). The Nash bargaining solution is f(f, v) = (4.5, 5.5). Choice 3: Rational Threats The threat game Γ derived from Γ is as follows. a b a 0, 0, 8 b 7.5,.5 0, 0 The unique equilibrium of this threat game = (a,b ) which leads to v = ( 5,) f(f,v) = (,8) In all the three cases, player chooses b in any disagreement because a is dominated by b with respect to both player s defensive objective of maximizing u player s offensive objective of minimizing u Player s disagreement behavior is different across the three cases. In case (Nash equilibrium case), player s behavior in the event of disagreement would be determined by his purely defensive objective of maximizing u. So he chooses b allowing player to get his maximum payoff In case (minimax case), player is supposed to be able to select between two threats: an offensive threat a for determining v a defensive threat b for determining v. 4

25 In case 3 (rational threats case), player must choose a single threat that must serve both offensive and defensive purposes simultaneously, so he chooses a because a maximizes the objective 0 + u u that is a synthesis of his offensive and defensive criteria. Which of these three methods is appropriate?. Equilibrium theory of disagreement is appropriate where the players could not commit themselves to any planned strategies in the event of disagreement, until a disagreement actually occurs.. The rational threats theory is applicable when each player can, before the negotiation or arbitration process, commit himself to a single planned strategy that he would carry out in the event of disagreement, no matter whose final rejection may have caused the disagreement. It is implicitly assumed that the probability of a disagreement is very very low. 3. The minimax values theory is appropriate in situations where each player can, before the process starts, commit himself to two planned strategies. (a) defensive (b) offensive He would carry out one of these depending on how the disagreement was caused. We can suppose that he would implement his defensive strategy if he himself rejected the last offer in the negotiation or arbitration process and his offensive strategy otherwise. To Probe Further The original discussion of the Nash bargaining problem and its solution is of course found in the classic paper of Nash []. The discussion and approach presented in this chapter closely follows that of Myerson [3]. There are two books that deal with the bargaining problem quite extensively. The book by Abhinay Muthoo [4] deals exclusively with the bargaining problem. The book by Osborne and Rubinstein [5] discusses bargaining theory extensively and presents rigorous applications to market situations of different kinds. The book by Straffin [6] provides two real-world applications of the two person bargaining problem: () Management - labor union negotiations, and () Duopoly Model of two competing companies trying to maximize their revenues. The concept of Nash program is discussed by Nash []. Problems. Suppose F is the convex hull enclosing the points A= (,8); B = (6,7); C = (8,6); D = (9,5); E =(0,3); F = (, -); and G = (-,-). Suppose the default point (status quo point) is (,). Compute the Nash bargaining solution for this situation. Write down a picture and that will be helpful.. Consider the following two player game: 5

26 A B A 5, 5, B, 3,3 For this game, compute the set of all payoff utility pairs possible (a) under correlated strategies (b) under individually rational correlated strategies. What would be the Nash bargaining solution in cases (a) and case (b) assuming the minmax values as the disagreement point. 3. Consider the following situation. It is required to lease a certain quantity of telecom bandwidth from Bangalore to Delhi. There are two service providers in the game {, }. Provider offers a direct service from Bangalore to New Delhi with a bid of 00 million rupees. Provider also offers a service from Mumbai to Delhi with a bid of 30 million rupees. On the other hand, Provider offers a a direct service between Bangalore and Delhi with a bid of 0 million dollars and a service from Bangalore to Mumbai with a bid of 50 million rupees. Compute a Nash bargaining solution assuming (0, 0) as the disagreement point. What can you infer from the Nash bargaining solution? References [] John F. Nash Jr. The bargaining problem. Econometrica, 8:55 6, 950. [] John F. Nash Jr. Two person cooperative games. Econometrica, :8 40, 953. [3] Roger B. Myerson. Game Theory: Analysis of Conflict. Harvard University Press, Cambridge, Massachusetts, USA, 997. [4] Abhinay Muthoo. Bargaining Theory with Applications. Cambridge University Press, 999. [5] Martin J. Osborne and Ariel Rubinstein. Bargaining and Markets. Academic Press, 990. [6] Philip D. Straffin Jr. Game Theory and Strategy. The Mathematical Association of America,