Holding Out for Concession: The Quest for Gain in the Negotiation of International Agreements

Abstract

If the terms of a potential international agreement are interpreted as sharing surpluses generated by cooperation, they can be viewed, on an item by item basis, as giving an advantage to one or the other party. By interpreting bargaining outcomes in terms of gains or losses relative to the status quo, we modify Rubinstein's full information bilateral alternative offers bargaining model, and find that this opens up new bargaining tactics in equilibrium. The Rubinstein solution no longer provides unique resolution to the bargaining problem. Instead, negotiators can choose from a continuum of strategies that involve holding tough positions and waiting for the other to concede. The tactic is visible in negotiations as diverse as Strategic Arms Limitation Talks (SALT) or North American Foreign Trade Agreement (NAFTA). This occurs although our negotiators are assumed to be fully informed about each other's priorities. Given the other's offer, a negotiator can choose optimally from a whole range of counteroffers in our model. But all of these leave her with the same ex ante expected utility. By focusing on the possible outcomes of a given strategy choice, we define an expected ex post evaluation of strategy that discriminates between strategies that were equivalent ex ante. Negotiators who seek to maximize this ex post valuation choose a tough bargaining position, offering less than the Rubinstein share, and wait, hoping that the other side will concede before they do.

Keywords:

• game theory
• bargaining
• s-shaped utility
• inefficiency

Notes

*The listing order of the authors’ names is not indicative of their respective contributions, which they consider to be equal.

¹ Kennan and Wilson (1993) and Muthoo (1999) provide extensive reviews of this literature.

² Sakovics (1993) shows that delays in complete information bargaining games can occur if moves are simultaneous. Cai (2000b,2003) shows how inefficient delays get introduced in a multilateral complete information bargaining game. Fernandz and Glazer (1991) show that costly strikes can occur despite the fact that labor and management are fully informed about each other's priorities. Slantchev (2003) shows that fully informed rivals can engage in costly wars despite the possibility of bargaining. Finally, A growing body of literature in political science focus on inefficient equilibria in stochastic games. The inefficiency results from the inability of the fully informed interested parties to commit to a strategy in changing times: For example,Powell (1999) examines a declining power's incentive to fight while Fearon (2002) and Acemoglu and Robinson (2001) analyze the incentives of revolutionary groups to challenge the elite in power.

³ Prospect theory and its application to international relations presents a range of possibilities and challenge, and we do not pretend, here, to incorporate the ideas of prospect theory in all their complexity. In particular, the framing of outcomes in terms of losses or gains poses a number of subtle problems that we do not address. In the interpretation that we propose, we are able to “subsume loss aversion and the reflection effect within expected utility theory by positing an S-shaped utility function” (Levy, 1997, p. 92) because we assume that the current status quo of relative power is the reference point. We do not incorporate the dynamic aspects of framing that would suggest that the reference point itself varies as negotiations progress. Neither do we give value to the reference point itself by interpreting it as being in the domain of losses or the domain of gains, an analysis that leads to an interpretation of state perpectives on possible movements away from the status quo (Berejekian, 1997).

⁴ Pre-multiplying the per period cost by (1−ω) simplifies calculations. We also assume symmetric costs for simplicity.

⁵ Note that because we assume for simplicity that both parties have the same utility function, if for negotiator 1, it is also for negotiator 2. In such a case any bargained outcome that would leave each party with less than would put them both in the “domain of losses.” This would be a bargain that neither party is truly happy with, each party believing that she has made too many concessions.

⁶ Rubinstein's actual argument is considerably more sophisticated than the above outline would suggest and is free of assumptions of symmetry or stationarity of offers.

⁷ If one alternative was strictly better than the other it would rationally be chosen with certainty, thus excluding probabilistic behavior.

⁸ Point R has coordinates (r, r),r being the Rubinstein offer.

⁹ There is a great deal of flexibility on the contingencies outside that are compatible with an SPE. In Player 1 could offer more than x = 0, and A could be subdivided so that a paltry offer y in response to overly generous x would never be accepted.

¹⁰ Among the possible picks is point that would be instantly accepted since p = q = 1. So, the Rubinstein solution appears here as a particular case.

¹¹ In a typical argument the Rubinstein offer is then shown to approach the Nash solution.

¹² On the boundary of one of the two probabilities is 1 except at point where they are both equal to 1.

¹³ Interestingly an evolutionary approach to equilibrium selection following authors such as Binmore, Piccione, and Samuelson (1998) would require that strategies be assessed according to the resulting utilities to each party. This suggests that the outcome selected by our ex post criterion could also emerge from an evolutionary analysis.

¹⁴ At node M, Player 1 may consider path τ₁ = ME (E for exit) which means immediate acceptance of offer y for a utility v(τ₁) = u(y) with probability π(τ₁) = p. All other r paths begin with move MN and have total probability (1 − p) with Player 2's likelihood of accepting offer x determining Player 1's ex ante indifference between the immediate acceptance of y and reiteration of offer x. Among the paths that can be taken if Player 1 does not accept y immediately, τ₂ = MNE yields utility (−(1 − w)c + wu(1 − x)) with probability π(τ₂) = (1 − p)q. All further paths τ _k would involve traveling MNM followed by all possible paths from M. Formula (2), when pre-multiplied by (1 − p) is therefore nothing but the expected utility of all paths from M except τ₁.

¹⁵ If one of the offers is immediately accepted at S, the bargain z is immediately reached. But since it has taken no time θ(τ) = 0 the ex ante and ex post values of that path coincide.

¹⁶ It is important to understand why this measure is distinct from the standard ex ante expected utility V. In V each possible endgame value is implicitly weighed by the discount factor w ⁿ corresponding to the turn n = θ(τ) at which the bargain is reached. This is because V is a view from the start. In W _w all endgames are treated equally regardless of their date of occurrence due to the factors w ^−θ(τ). This is because W _w is a view from the end.

¹⁷ The break in slope at reflects the change in the functional form of W. For it is u(1 −x).

¹⁸ Congressional Record, 1867, as quoted by Barker (2002).

¹⁹ http://www.virtualology.com/virtualpubliclibrary/hallofthehistoricarchives/ALASKAPURCHASE.COM/

²⁰ While our model can incorporate the sequence of offers and counteroffers that lead up to final bargaining positions, we are interested in the characterization of the final demands and focus exclusively on these in the modeling effort as well as the discussion of real world negotiations.

²¹ The so-called proportional access demanded by the U.S. would have committed Mexico to limit the possible decrease in oil exports to the U.S. in the times of crisis.

²² According to Lowenthal (1982) the maximum weight of nuclear warheads and reentry vehicles that a missile could carry (throw-weight) was estimated at 2,220,330 pounds for the U.S. and 9,954,100 pounds for the Soviet Union.

²³ The definition of “heavy” remained unresolved. According to Smith (1980, pp. 334), the Soviets agreed that SS-9s were heavy while SS-11s were “light,” but the SS-9 was three times as large as the SS-11 and the delegates were not able to agree on a cutoff point between these two extreme missile sizes.

²⁴ The bargaining theories developed by authors such as Zeuthen (1930) and Cross (1969) also emphasize concession in their bargaining theories, but these authors examine concession rates and the convergence to an agreement “defined by a situation in which the sum of the players’ demands is equal to available supply” (Cross, 1965, p. 81). The holding-out behavior that we identify leads to a bargaining solution in which one party gives in to the others’ demand in a situation where the sum of outstanding demands exceeds supply. Moreover, as Wagner (1979) points out, if the negotiators are well informed of each other's priorities, Zeuthen and Cross's arguments cannot lead negotiators to incur costly delays in the hope that the other will concede.

²⁵ ξ is defined for all 0 ≤ z ≤ u ⁻¹(w(1 −k)) and η is defined for all 0 ≤ z ≤ 1 − u ⁻¹ (k). Assumption I and the monotonicity of u imply that r < min {k,w(1 −k)}.

²⁶ Namely, for any t ∈ {3,5,7,..} (i) if (x _t , y _t−1) ∈ then Player 2 accepts x _t with probability q = q (x _t , y _t−1) given by (4) but offers y _t+1 = y _t−1 with probability (1 − q): (ii) If (x _t , y _t−1) ∈ then he accepts x _t; (iii) If (x _t , y _t−1) ∈ C ₁ then he offers y _t+1 = 0.

²⁷ Specifically:

Barker, David (2002). “Was the Alaska Purchase a Good Deal?” Manuscript, University of Chicago Graduate School of Business.

Brams, Steven, Marc Kilgour, and M. Remzi Sanver (2003). “A Minimax Procedure for Negotiating Multilateral Treaties,” unpublished manuscript.

Fearon, James D. (2002). “Why Do Some Civil Wars Last So Long?” Manuscript, Department of Political Science, Stanford University.

Lowenthal, Mark M. (1982). “The START Proposal: Verification Issues.” Congressional Research Service, June 25, p. 1, Washington DC.