Bargaining problem

The negotiated settlement is a game theoretical approach to the solution of cooperative games. This game is called a cooperative if the players can achieve a concerted approach, ie, by a common choice of a strategy, an additional gain compared to the situation in which everyone plays only for himself. In this case, to negotiate the allocation of the additional profit, hence the term negotiated solution ( engl. bargaining solution). This item is limited to so-called two-person games, ie games in which only two players are involved.

• 7.1 The Nash'sche negotiated solution
• 7.2 The monotonic bargaining solution

The non-cooperative situation

Most games require frequent strategic choices of the players involved to bring about a favorable outcome of the game for them. Based on these decisions for all possible game situations before the game begins firmly, one has to deal only with a strategy for each player. The execution of the game is then only in compliance with the decisions already taken. This is the view of the mathematician:

A non- cooperative two-person game consists of two volumes and and two pictures, one short writes. The game is that each player chooses independently of the other an element of its strategy set. The i-th player scores then the payout.

If the strategy quantities last, so you can number them and put about. The payoff functions are then given by two matrices and one speaks of a bimatrix game.

Each player can guarantee a certain payment amount by choosing the best strategy for imputing the worst for him strategy choice of an opponent

• Player 1:
• Player 2:.

These are the so-called guaranteed values ​​of the players. Become supremum and infimum is not accepted, one has nevertheless still approximate guarantees.

Under a Nash equilibrium will be understood from a pair of strategies, so that a player can deteriorate through unilateral departure from its strategy at most. In the theory of non-cooperative games, in which a common collusive deviation is not provided, a balance can be understood as a solution of the game.

The Prisoner's Dilemma

The most well-known game-theoretic situation that absolutely challenges the negotiation term, is the prisoner's dilemma. Two prisoners are accused on the basis of dubious evidence of a common crime. Each has two strategies to choose from: 1 = deny, 2 = Confess. Denying both, only a one-year prison sentence may be imposed, such as for illegal weapons possession and disturbing the peace. Confess both, so each 8 years overdue. Meet different strategies, the confessor will go unpunished as a witness, the deniers, however, faces a 10 - year prison over. Taking as payment in this bimatrix game the negative of the abzusitzenden years imprisonment, shall apply to the payoff

The only equilibrium point is obviously (2.2 ), that is, admit both prisoners. ( 1.1 ) is not an equilibrium because each player can secure by a change of strategy at the expense of other impunity. Since everyone involved knows the strategy combination (1,1) appears even very unstable.

Nevertheless, ( 1.1 ) is certainly the best solution from the perspective of delinquents. But for that they need to make an appointment, that is, negotiate the strategies to be employed. This is modeled by the concept of cooperation.

Cooperation

In order to model mathematically negotiations, we extend the definition of the non- cooperative game with and to. We call K the set of cooperative strategies. When choosing a cooperative strategy k is the ith player gets the payout. Since each player can also play its own strategy, but these are possibly due to an agreement with the player who is now no longer regarded as an opponent.

The Prisoner's Dilemma, we can model with it then just comes down to the possibility of collusion. Looking at about two economic actors who own production strategies for the same market in the non-cooperative situation, cooperative strategies are conceivable, which goes beyond an agreement of production strategies, such as the creation of a cartel, or the coordination of production by cascading various processing stages. What is allowed as a cooperative strategy is the content of the rules, the antitrust law is a rule of the game.

Negotiation situations

We focus now on the image of the common payoff function in, ie we abstract from the strategies that lead to these payouts. B contains a point that is made payouts that each player also can alone secure, such as defined above guaranteed values ​​. Players will certainly negotiate only on payouts, where and is, otherwise, a player would be better off with its guaranteed value. Furthermore, should a possible payout with and exist so that there is for both players to negotiate anything.

In addition, we allow to choose a joint probability distribution on K the players. The payout point is then calculated as the expected value. The set B of the possible payout points, we may therefore assume as a convex, because by a common choice of players can realize any convex combination of payout points. Furthermore, we can assume that B is limited by excluding unlimited payoff functions as unrealistic. Suppose B also still considered complete, B is even compact. This motivates the following going back to Ehud Kalai and Meir Smorodinsky concept formation:

A bargaining situation is a pair with the following properties:

• Convex and compact,
• ,
• Applies to all component-wise,
• There is a with and.

The transaction problem

If a negotiation situation, it is called the choice of a payment point a negotiation result. The i-th player receives the payoff. The bargaining problem is to be found in any negotiation situation, such a negotiation result. Is the set of all bargaining situations, we therefore define:

A bargaining solution is a function with for all.

Properties of negotiated solutions

Of course, you will require certain properties of a negotiated solution that makes the solution appear to be " reasonable." Thus, the negotiated settlement of all is certainly not very "reasonable" because no player gets more than he could secure in any case by the trial. It is therefore in the following is to find meaningful properties, with the aim of thereby define a unique payout point in any negotiation situation.

Pareto optimality

A negotiated solution is called Pareto optimal if there is one for any negotiation situation with component-wise. That is, there is always found a negotiated solution that does not allow simultaneous betterment of both parties.

This condition is mathematically plausible. In practice it may be difficult to find such not be improved upon negotiated solutions.

Symmetry

A negotiated solution is called symmetric if the following applies: If the negotiation situation symmetrical, ie and for all is also so agree, the components of the negotiation result.

Thus, it is required that the negotiated solution does not change in a completely symmetric situation when the players switch roles. Both players will be subject to the same negotiating skills.

Independence of positive linear transformations

We consider with positive linear transformations. T is for both components a scale change along with a shift. A negotiated solution is independent of positive linear transformations, if T and any negotiation situation is true for any positive linear transformation that.

The requirement for independence of positive linear transformations is mathematically very obvious and for many mathematical considerations also essential. In practice, this means that the negotiations are independent before the scale size of the bargaining chips. Since negotiations take time and resources, you may raise concerns for the practical relevance of this requirement.

Independence of irrelevant alternatives

A negotiated solution is independent of irrelevant alternatives, if for two Verhandlungssitationen with always.

This requirement seems obvious, said they, however, that a solution found in the larger transaction amount, which is already available in the smaller bargaining set B will also be the solution for the smaller transaction amount, because even in the larger transaction amount can not find anything better. Against this requirement can probably only psychological objections are: A change negotiation situation changes the negotiation behavior.

Monotony

For a negotiation was the maximum payment that would be at all possible for the i-th player. A negotiated solution is called monotonic if = 1.2 and always componentwise follows for with i.

Thus, if both players can back up only the amount 0 and both can achieve maximum payout 1, as should be apparent to any player deterioration, when one passes while maintaining these conditions from a smaller to a larger transaction amount.

Existence and uniqueness theorems

The Nash'sche negotiated solution

Set of Nash: There is exactly one Pareto - optimal, symmetric, independent of positive line rare transformations and independent of irrelevant alternatives negotiated solution.

If two players agreed to these here four exposures to a negotiated solution, then there is thus in every negotiation situation a unique bargaining solution, this is called the Nash'sche negotiated solution. This negotiated settlement can be determined as follows: If a negotiation situation, as assumes the function in exactly one point from B, the maximum, and this point is the Nash'sche negotiated solution.

The Nash'sche negotiated solution is not monotonic!

The monotonic bargaining solution

Kalai - Smorodinsky set of: There is exactly one Pareto - optimal, symmetric, independent of positive linear transformations and monotonous negotiated solution.

This solution is called the monotone negotiated solution. To determine the monotonic bargaining solution is determined at a given negotiation situation a positive linear transformation T such that and. In a straight line, there is a respect to the componentwise order biggest point. The searched result of the negotiations is then.

Negotiated solution to the prisoner's dilemma

In the situation of the prisoner's dilemma is the set of possible joint payoffs with the above numbers from the four points (-8, -8 ), ( 0, -10 ), ( -10,0 ) and ( -1, -1). The convex hull is the quadrilateral formed by these points. The warranty period is d = (-8, -8 ). About not gray shaded points adjacent drawing, there is nothing to negotiate, the gray shaded area is therefore the transaction amount B. Both solution concepts that Nash'sche and the monotonous negotiated solution leading to (-1, -1) as a solution.

Concluding Remarks

• Frequently we read of the requirement of individual rationality, according to which the outcome of negotiations may be componentwise no worse than the guarantee point always. In the given representation here, this requirement lies in the definition of the negotiation situation.
• To monotony you could have claimed more generally, that can only improve the situation for both players in any negotiation situation, if you increase the bargaining set B under the same conditions. It turns out that this requirement is so strong that it does not permit more bargaining solution.
• In the monotonous negotiated solution over the Nash'schen negotiated settlement only the independence of irrelevant alternatives has been replaced by the monotony requirement.
• The Nash'sche and the monotonous negotiated solution coincide in symmetric bargaining situations such as the prisoner's dilemma.