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The Two-Person Bargaining Problem

The two person bargaining problem is a problem of understanding how two agents should cooperate when non-cooperation leads to Pareto-inefficient results. It is in essence an equilibrium selection problem. Many games have multiple equilibria with varying payoffs for each player, forcing the players to negotiate on which equilibrium to target. The quintessential example of such a game is the Ultimatum game. The underlying assumption of bargaining theory is that the resulting solution should be the same solution an impartial arbitrator would recommend. Solutions to bargaining come in two flavors: an axiomatic approach where desired properties of a solution are satisfied and a strategic approach where the bargaining procedure is modeled in detail as a sequential game.

An example
The Battle of the Sexes, as shown, is a two player coordination game. Both Opera/Opera and Football/Football are Nash equilibria. Any probability distribution over these two Nash equilibria is a correlated equilibrium. The question then becomes which of the infinite possible equilibria should be chosen by the two players. If they disagree and choose different distributions then they will fail to coordinate and likely receive 0 payoffs. In this symmetric case the nature choice is to play Opera/Opera and Football/Football with even probability. Indeed all bargaining solutions described below prescribe this solution. However if the game is asymmetric (for example Football/Football instead yields payoffs of 2,5) the appropriate distribution becomes less clear. Bargaining theory solves this problem.

The Formal Description
A 2 person bargain problem consists of a disagreement point $$v$$ (also known as a threat point) and a feasibility set $$F$$. $$v = (v_1, v_2)$$, where $$v_1$$ and $$v_2$$ are the payoffs after disagreement to player 1 and player 2 respectively. $$F$$ is a closed convex subset of $$\textbf{R}^2$$ representing the set of possible agreements. $$F$$ is convex because an agreement could take the form of a correlated combination of other agreements. Points in $$F$$ must all be better than the disagreement point as there is no sense to an agreement which is worse than disagreement. The goal of bargaining is to choose the feasible agreement $$\phi$$ in $$F$$ that would result after thorough negotiations.

Feasibility Set
The set of possible agreements $$F$$ depends on if there is an outside regulator affording binding contracts. When binding contracts are allowed any joint action is playable so the feasibility set consists of all attainable payoffs better than the disagreement point. When binding contracts are not allowed the game is said to have moral hazard (as players can defect) and thus the feasibility set only consists of correlated equilibrium, which need no enforcement.

Disagreement Point
The disagreement point $$v$$ is the value the players can expect to receive if negations break down and no bargain can be reached. Naively this could just be some focal equilibrium to which both players could expect to play. However, this point directly affects eventual bargaining solution, so it stands to reason that each player should attempt to choose their disagreement points in order to maximize their bargaining position. Towards this goal, it is often advantageous to simultaneously increase one’s own disagreement payoff while harming one’s opponent's disagreement payoff - hence this point is often known as the threat point. If threats are viewed as actions then we can construct a separate game where each player chooses a threat and receives a payoff according to the outcome of bargaining. This is known as Nash’s variable threat game. Alternatively each player could play a minimax strategy in case of disagreement, choosing to disregard personal reward in order to hurt the opponent as much as possible if they leave the bargaining table.

Nash bargaining solution
On the other hand, Nash proposed that a solution should satisfy certain axioms, 1) Invariant to affine transformations or Invariant to equivalent utility representations, 2) Pareto optimality, 3) Independence of irrelevant alternatives, 4) Symmetry. Let us call u the utility function for player 1, v the utility function for player 2. Under these conditions, rational agents will choose what is known as the Nash bargaining solution. Namely, they will seek to maximize $$|u(x)-u(d)||v(y)-v(d)|$$, where $$u(d)$$ and $$v(d)$$, are the status quo utilities (i.e. the utility obtained if one decides not to bargain with the other player). The product of the two excess utilities is generally referred to as the Nash product.

Kalai-Smorodinsky bargaining solution
Independence of Irrelevant Alternatives can be substituted with an appropriate monotonicity condition, thus providing a different solution for the class of bargaining problems. This alternative solution has been introduced by E. Kalai and M. Smorodinsky. It is the point which maintains the ratios of maximal gains. In other words, if player1 could receive a maximum of g_1 with player2’s help (and visa-versa for g_2), then the Kalai-Smorodinsky bargaining solution would yield the point $$\phi$$ on the Pareto frontier such that $$\phi_1 / \phi_2 = g_1 / g_2  $$.

Egalitarian bargaining solution
The egalitarian bargaining solution is a third solution which drops the condition of scale invariance while including both the axiom of Independence of irrelevant alternatives, and the axiom of monotonicity. It is the solution which attempt to grant equal gain to both parties.

Applications
Recently the Nash bargaining game has been used by some philosophers and economists in order to explain the emergence of human attitudes toward distributive justice (Alexander 2000; Alexander and Skyrms 1999; Binmore 1998, 2005). These authors primarily use evolutionary game theory in order to explain how individuals come to believe that proposing a 50-50 split is the only just solution to the Nash Bargaining Game.