Abstract
I study the following repeated version of Nash’s Demand Game: whenever the demands are not jointly compatible, the player who stated the lower demand (the less greedy player) obtains the following advantage: his offer is the only one “on the table”, and the greedier player needs to respond to this offer by either accepting it (which terminates the game) or rejecting it (which triggers a one-period delay and a re-start of the game). If the feasible set is regular—meaning that the egalitarian point is also utilitarian—the game has a unique subgame perfect equilibrium. The equilibrium outcome is an immediate agreement on the egalitarian point. Regularity of a feasible set is a weakening of symmetry. Under some equilibrium refinement, regularity can be dispensed with.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
These are common assumptions in the NDG literature.
In an earlier work, Binmore (1987) showed that Nash’s suggestion holds for a certain class of parametrized perturbations.
An alternative smoothing-technique is to apply perturbations not to the feasible set, but to the players’ strategies. This possibility has been explored by Carlsson (1991), who added a noise component to the players’ demands.
It is easy to check that the constant strategy under which player 1 always demands x and player 2 always demands y, where (x, y) is Pareto efficient, can be made part of a subgame perfect equilibrium. In order to complete the strategy, one needs to specify off-path behavior, namely to specify responses to proposals that should not have occurred on the path. Let player i accept an offer if and only if his payoff under this offer is at least \(\delta _i\) times his share in (x, y), where \(\delta _i\) is his discount factor.
More precisely, if a player agrees to the payoff u after some particular history h, then he also agrees to accept any \(u'>u\) after this particular history.
Additionally, under monotone acceptance any tie-breaking rule works, while without monotone acceptance symmetric tie-breaking is assumed.
Anbarcı et al. (2018) discuss this towards the end of their Introduction.
The analogy with Bertrand is as follows: in Bertrand competition it only matters who offers the lowest price, the prices themselves do not matter for the winner’s determination; here, analogously, it only matters who is the less greedy player, the offers themselves do not matter for the proposer’s determination.
Malueg (2010) studies mixed-strategy equilibria in NDG and shows how disagreement can arise in equilibrium. Güth et al. (2004) study a perturbed NDG in which the players have an option to wait until the uncertainty (the perturbation of the feasible set) is resolved before they state their demands. They show that the strict equilibria of this game are asymmetric. An equilibrium is non-strict if every action is a best-response. For example, given a strictly decreasing Pareto frontier, a non-strict equilibrium obtains in NDG when each player demands his maximum possible utility (this is true both in the original NDG and in the version of Güth et al.).
In Section 6 (and only there) I allow S to be non-convex.
Vector inequalities: uRv if and only if \(u_i R v_i\) for both \(i\in \{1,2\}\), for both \(R\in \{>,\ge \}\); \(u\gneqq v\) if and only if \(u\ge v\) and \(u\ne v\).
If the stage game is such that (i) player 2’s payoff is \(u_2\) and (ii) there is certain disagreement and move to the next period, then clearly there is some other stage game down the game-tree in which (i’) player 2’s payoff is at least \(u_2\) and (ii’) the probability of acceptance is positive. Note that the acceptance probability can be strictly between zero and one despite the fact that strategies are pure, because there is the possibility that in case of a tie only one player accepts the offer while the other does not.
\(g(t)=t\) for \(t=e\) and g is decreasing. For a small \(\epsilon '>0\) it holds that \(M_2-\epsilon -\epsilon '>\delta _2M_2\ge \delta _2C_2\ge e\) and this implies \(x'=g(M_2-\epsilon -\epsilon ')<e\le \delta _2M_2<y\).
This follows from regularity: \(x+f(x)\le \text {max}_a[a+f(a)]=2e\).
\(x>e\) implies \(f(x)<e\le \delta _2M_2\).
I deliberately use here the general-\(\lambda \)-notation, despite the fact that the theorem is for \(\lambda =\frac{1}{2}\), because this general formulation will shortly be handy, in Corollary1 below.
This is because \(\frac{1}{1-\lambda }[M_2(1-\lambda \delta _2)-\epsilon ]>\delta _2M_2\), which follows from (1).
\(f(x')=\delta _2M_2\ge e\) implies \(x'\le e\), and \(e<x\).
Because S is convex, every pair of expected utilities—even utilities the generation of which involves randomization (due to random proposer-selection) and/or delays—are elements of S, and \((e,e)\in P(S)\).
Given \(V\subset {\mathbb {R}}_+^2\), \(\text {comp}V\) is the smallest comprehensive set (in \({\mathbb {R}}_+^2\)) that contains V.
This is an absorbing state in which play stays forever.
By (I), rejection triggers the egalitarian equilibrium as the continuation equilibrium.
This follows from (III).
Note that, according to (III), rejection by player 2 implies that player 1’s deviation is ignored, and play starts fresh in the next period as if nothing happened.
If, for example, player 1 deviates in the first period and demands e instead of a, then player 2 would prefer to wait and obtain (in expectation) the payoff \(\frac{a+e-\epsilon }{2}\) tomorrow, rather than settle for e today.
I thank an anonymous referee for pointing this out.
If the demands coincide, the responder is selected at random.
For brevity, I skip the details. Also, it is straightforward that an equilibrium of the static game must have the monotone acceptance property.
References
Abreu D, Pearce D (2015) A dynamic reinterpretation of Nash bargaining with endogenous threats. Econometrica 83:1641–1655
Anbarcı N (2001) Divide-the-dollar game revisited. Theory Decis 50:295–304
Anbarcı N, Boyd JH (2011) Nash demand game and the Kalai–Smorodinsky solution. Games Econ Behav 71:14–22
Anbarcı N, Rong K, Roy J (2018) Random-settlement arbitration and the generalized Nash solution: one-shot and infinite-horizon cases. Econ Theor 68:21–52
Ashlagi I, Karagözoğlu E, Klaus B (2012) A non-cooperative support for equal division in estate division problems. Math Soc Sci 63:228–233
Binmore K (1987) Nash bargaining theory (II). In: Binmore K, Dasgupta P (eds) The economics of bargaining. Basil Blackwell, Cambridge
Brams SJ, Taylor AD (1994) Divide the dollar: three solutions and extensions. Theor Decis 37:211–231
Carlsson H (1991) A bargaining model where parties make errors. Econometrica 59:1487–1496
Cetemen ED, Karagözoğlu E (2014) Implementing equal division with an ultimatum threat. Theor Decis 77:223–236
Dutta R (2012) Bargaining with revoking costs. Games Econ Behav 74:144–153
Güth W, Ritzberger K, Van Damme E (2004) On the Nash bargaining solution with noise. Eur Econ Rev 48:697–713
Howard JV (1992) A social choice rule and its implementation in perfect equilibrium. J Econ Theory 56:142–159
Kambe S (1999) When is there a unique equilibrium in less structured bargaining? Jpn Econ Rev 50:321–342
Karagözoğlu E, Rachmilevitch S (2018) Implementing egalitarianism in a class of Nash demand games. Theor Decis 85:495–508
Malueg DA (2010) Mixed-strategy equilibria in the Nash Demand Game. Econ Theor 44:243–270
Muthoo A (1996) A bargaining model based on the commitment tactic. J Econ Theory 69:134–152
Nash JF Jr (1950) The bargaining problem. Econometrica 18:155–162
Nash JF (1953) Two person cooperative games. Econometrica 21:128–140
Rachmilevitch S (2015) The Nash solution is more utilitarian than egalitarian. Theor Decis 79:463–478
Rachmilevitch S (2017) Punishing greediness in divide-the-dollar games. Theor Decis 82:341–351
Rachmilevitch S (2019a) Folk theorems in a bargaining game with endogenous protocol. Theor Decis 86:389–399
Rachmilevitch S (2019b) Alternating offers bargaining and the golden ratio. Fibonacci Q 57:299–302
Rubinstein A (1982) Perfect equilibrium in a bargaining model. Econometrica 50:97–109
Shapley L (1969) Utility comparison and the theory of games, in“La Decision” (G. Guilbaud, Ed.). Editions du CNRS, Paris
Acknowledgements
Reports of several anonymous referees are greatly appreciated.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Rachmilevitch, S. Rewarding moderate behavior in a dynamic Nash Demand Game. Int J Game Theory 49, 639–650 (2020). https://doi.org/10.1007/s00182-019-00704-1
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00182-019-00704-1