“The Nash Bargaining Solution in Economic Modeling,” K. Binmore, A. Rubinsten & A. Wolinsky (1986)

If we form a joint venture, our two firms will jointly earn a profit of N dollars. If our two countries agree to this costly treaty, total world welfare will increase by the equivalent of N dollars. How should we split the profit in the joint venture case, or the costs in the case of the treaty? There are two main ways of thinking about this problem: the static bargaining approach developed first by John Nash, and bargaining outcomes that form the perfect outcome of a strategic game, for which Rubinstein (1982) really opened the field.

The Nash solution says the following. Let us have some pie of size 1 to divide. Let each of us have a threat point, S1 and S2. Then if certain axioms are followed (symmetry, invariance to unimportant transformations of the utility function, Pareto optimality and something called the IIA condition), the bargain is the one that maximizes (u1(p)-u1(S1))*(u2(1-p)-u2(S2)), where p is the share of the pie of size 1 that accrues to player 1. So if we both have linear utility, player 1 can leave and collect .3, and player 2 can leave and collect 0, but a total of 1 is earned by our joint venture, the Nash bargaining solution is the p that maximizes (p-.3)*(1-p-0); that is, p=.65. This is pretty intuitive: 1-.3-0=.7 of surplus is generated by the joint venture, and we each get our outside option plus half of that surplus.

The static outcome is not very compelling, however, as Tom Schelling long ago pointed out. In particular, the outside option looks like a noncredible threat: If player 2 refused to offer player 1 more than .31, then Player 1 would accept given his outside option is only .3. That is, in a one-shot bargaining game, any p between .3 and 1 looks like an equilibrium. It is also not totally clear how we should interpret the utility functions u1 and u2, and the threat points S1 and S2.

Rubinstein bargaining began to fix this. Let players make offers back and forth, and let there be a time period D between each offer. If no agreement is reached after T periods, we both get our outside options. Under some pretty compelling axioms, there is a unique perfect equilibrium whereby player 1 gets p* if he makes the first offer, and p** if player 2 makes the first offer. Roughly, if the time between offers is D, player 1 must offer player 2 a high enough share that player 2 is indifferent between that share today and the amount he could earn when he makes an offer in the next period. Note that the outside options do not come into play unless, say, player 1’s outside option is higher than min{p*,p**}. Note also that as D goes to 0, all of the difference in bargaining power has to do with who is more patient. Binmore et al modify this game so that, instead of discounting the future, rather there is a small chance that the gains from negotiation will disappear (“breakdown”) in between every period; for instance, we may want to form a joint venture to invent some product, but while we negotiate, another firm may swoop in and invent it. It turns out that this model, with von Neumann-Morganstern utility functions for each player (though perhaps differing levels of risk aversion) is a special case of Rubinstein bargaining.

Binmore et al prove that as D goes to zero, both strategic cases above have unique perfect equilibria equal to a Nash bargaining solution. But a Nash solution for what utility functions and threat points? The Rubinstein game limits to Nash bargaining where the difference in utilities has to do with time preference, and the threat points S1 and S2 are equal to zero. The breakdown game limits to Nash bargaining where the difference in utilities has to do with risk aversion, and the threat points S1 and S2 are equal to whatever utility we would get from the world after breakdown.

Two important points: first, it was well known that a concave transformation of a utility function leads to a worse outcome in Nash bargaining for that player. But we know from the previous paragraph that this concave transformation is equivalent to a more impatient Rubinstein bargainer: a concave transformation of the utilities in the Nash outcome has to do with changing the patience, not the risk aversion, of players. Second, Schelling was right when he argued that the Nash threat points involve noncredible threats. As long as players prefer their Rubinstein equilibrium outcome to their outside option, the outside option does not matter for the bargaining outcome. Take the example above where one player could leave the joint venture and still earn .3. The limit of Rubinstein bargaining is for each player to earn .5 from the joint venture, not .65 and .35. The fact that one player could leave the joint venture and still earn .3 is totally inconsequential to the negotiation, since the other player knows that this threat is not credible whenever the first player could earn at least .31 by staying. This point is often wildly misunderstood when people apply Nash bargaining solutions: properly defining the threat point matters!

Final RAND version (IDEAS). There has been substantial work since the 80s on the problem of bargaining, particularly in trying to construct models where delay is generated, since Rubinstein guarantees agreement immediately and real-world bargaining rarely ends in one step; unsurprisingly, these newer papers tend to rely on difficult manipulation of theorems using asymmetric information.

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