# Bargaining over Strategies of Non-Cooperative Games

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## Abstract

**:**

## 1. Introduction

- Player (i):
- “If I play strategy ${s}_{i}^{1}$, which strategy would you play?”.
- Player (j):
- “If you play ${s}_{i}^{1}$, I would play ${s}_{j}^{2}$”.
- Player (i):
- Either “Ok, I confirm that I’ll play strategy ${s}_{i}^{1}$, and so let us play (${s}_{i}^{1}$,${s}_{j}^{2}$)”, or “No, if you play ${s}_{j}^{2}$, I would play ${s}_{i}^{3}$”.

- Player (j):
- Either “Ok, I confirm that I’ll play strategy ${s}_{j}^{2}$, and so let us play (${s}_{i}^{3}$,${s}_{j}^{2}$)”, or “No, if you play ${s}_{i}^{3}$, I would play ${s}_{j}^{4}$.

## 2. The Bargaining Supergame

_{k}the finite strategy space for player k (with $k=i,j$) in the original non-cooperative game G. Player k’s set of possible proposals in the supergame with confirmed proposals CP(G) coincides with S

_{k}. As a consequence, the set of possible agreements in CP(G) coincides with the set of strategies of G, i.e., the product set ${S}_{i}\times {S}_{j}$ contains all the possible agreements of CP(G).

- Period 1.
- Player i proposes a strategy ${s}_{i}^{1}\in {S}_{i}$ to player j. Player i would actually play ${s}_{i}^{1}$ if (and only if) she would confirm this strategy after the counter-proposal of player j.
- Period 2.
- Player j proposes a strategy ${s}_{j}^{2}\in {S}_{j}$ to player i. This strategy would actually be played if (and only if) either i will confirm her previous strategy ${s}_{i}^{1}$ or j will confirm her proposal ${s}_{j}^{2}$ after the counter-proposal of player i.
- Period 3.
- Player i chooses whether or not to confirm her previous strategy ${s}_{i}^{1}$. If she confirms ${s}_{i}^{1}$, i.e., ${s}_{i}^{3}$ = ${s}_{i}^{1}$, then the bargaining process ends, through the sequence (${s}_{i}^{1}$,${s}_{j}^{2}$,${s}_{i}^{1}$), with the confirmed agreement (${s}_{i}^{1}$,${s}_{j}^{2}$), and the two players receive the payoffs corresponding to the strategy profile (${s}_{i}^{1}$,${s}_{j}^{2}$) in the original game G. If she does not confirm, i.e., she proposes a new strategy ${s}_{i}^{3}\ne {s}_{i}^{1}$, the bargaining process continues with ${s}_{j}^{2}$ as player j’s proposal and ${s}_{i}^{3}$ as player i’s counter-proposal to j’s proposal.
- Period 4.
- Player j chooses whether or not to confirm her previous strategy ${s}_{j}^{2}$. If she confirms ${s}_{j}^{2}$, i.e., ${s}_{j}^{4}={s}_{j}^{2}$, then the bargaining process ends, through the sequence (${s}_{j}^{2}$,${s}_{i}^{3}$,${s}_{j}^{2}$), with the confirmed agreement (${s}_{j}^{2}$,${s}_{i}^{3}$), and the two players receive the payoffs corresponding to the strategy profile (${s}_{i}^{3}$,${s}_{j}^{2}$) in the original game G. If she does not confirm, i.e., she proposes a new strategy ${s}_{j}^{4}\ne {s}_{j}^{2}$, the bargaining process continues with ${s}_{i}^{3}$ as player i’s proposal and ${s}_{j}^{4}$ as player j’s counter-proposal to i’s proposal. And so on and so forth.

_{k}for player k (with $k=i,j$) in the original non-cooperative game G.4 Let $\overline{H}$ be the set of all feasible histories h, where ${h}^{0}$ indicates the initial, empty history of CP(G), i.e., before period 1, and ${h}^{t}$ for $t=1,2,\dots $ indicates a feasible history before period $t+1$.

_{i}if the active player $k=i$, and S

_{j}otherwise.

_{k}satisfies stationarity, i.e., the preference between two agreements does not depend on time: if ${s}_{k}^{t-2}={s}_{k}^{t\prime -2}$, ${s}_{k}^{t-1}={s}_{k}^{t\prime -1}$, ${\tilde{s}}_{k}^{t-2}={\tilde{s}}_{k}^{{t}^{\prime}-2}$ and ${\tilde{s}}_{k}^{t-1}={\tilde{s}}_{k}^{{t}^{\prime}-1}$, then $f({s}_{k}^{t-2},{s}_{-k}^{t-1})$ $\succsim $

_{k}$f({\tilde{s}}_{k}^{t-2},{\tilde{s}}_{-k}^{t-1})$ if and only if $f({s}_{k}^{t\prime -2},{s}_{-k}^{t\prime -1})$ $\succsim $

_{k}$f({\tilde{s}}_{k}^{t\prime -2},{\tilde{s}}_{-k}^{t\prime -1})$ for all t ≠ t′, with $t,t\prime =3,4,\dots $ The assumption of stationarity of preferences means that a player’s preferences do not depend on period t of an agreement in CP(G), but only on the outcome of G due to this agreement.

_{k}$f({s}_{k}^{t\prime -2},{s}_{-k}^{t\prime -1})$ for all t < t′, with $t,t\prime =3,4,\dots $

## 3. General Results about the Equilibrium of the Bargaining Supergame

**Example 1: One equilibrium confirmed agreement.**Consider the two-player simultaneous game G in Figure 1. The set of strategies for player i and player j is, respectively, S

_{i}= {Superior, Inferior}, henceforth S

_{i}= {S, I}, and S

_{j}= {Left, Right}, henceforth S

_{j}= {L, R}. Figure 1, with a > b > c > d, shows, besides the simultaneous-move original game G, also all the possible agreements of CP(G), the bargaining supergame with confirmed proposals built on it.

**Example 2: No equilibrium confirmed agreement.**Our solution procedure does not always allow for an equilibrium of CP(G). Consider Figure 3. The original game G has only one Nash equilibrium in mixed strategies, with player one choosing strategy S with probability $p(S)=0.25$ and player two choosing strategy L with probability $q(L)=\mathrm{0.5.}$ This leads to expected payoffs V

_{i}= V

_{j}= 1.5. Both expected payoffs are larger than 1, the second-lowest possible payoff of G.

_{k}in the mixed-strategy Nash equilibrium of G by not bargaining through CP(G).

_{k}. In fact, if player i’s first proposal in period 1 is S, the history that results by taking into account weak dominance and backward induction is the initial history (S, R, I, L) repeated infinite times.6 If, instead, player i’s first proposal in period 1 is I, the history that results by taking into account weak dominance and backward induction is the initial history (I, L, S, R) repeated infinite times.

_{i}= V

_{j}= 1.5.

**Example 3: One equilibrium confirmed agreement that is not played.**It can be the case that, although CP(G) has an equilibrium confirmed agreement, no commitment to play G according to the equilibrium confirmed agreement of CP(G) is possible, since one of the two players would get a higher payoff by directly playing G, i.e., in the Nash equilibrium of G. This happens, for example, when the original game G is the Entry Game. In this two-stage game, player i (the potential entrant) chooses whether to Enter (E) or to Stay Out (S) of the market, with j (the incumbent) deciding whether to Accommodate (A) or to Fight(F) if the entrant decides to enter. The strategic form of the game in Figure 5, where x:= “x if E”, with $x=A,F$, and a > b > c > d, represents all the possible agreements of CP(G). Notice that the highest possible payoff for player i is b.

**Uniqueness of the equilibrium confirmed agreement.**If an equilibrium exists, we can introduce Proposition 1, concerning the uniqueness of the equilibrium confirmed agreement.

**Proposition 1.**If the equilibrium for CP(G) exists for a given first mover, and G is generic, then the equilibrium confirmed agreement is unique, hence players agree on a unique behavior in G. If G is not generic, then multiple confirmed agreements are possible, although being payoff-equivalent for at least one player.

_{k}$f({s}_{i}\prime ,{s}_{j}\prime )$ if ${s}_{i}\ne {s}_{i}\prime $ and/or ${s}_{j}\ne {s}_{j}\prime $ for $k=i,j$.

_{k}$f({{s}^{\prime}}_{i},{{s}^{\prime}}_{j})$ if ${s}_{i}\ne {{s}^{\prime}}_{i}$ and/or ${s}_{j}\ne {{s}^{\prime}}_{j}$ for $k=i,j$. Suppose that $f({s}_{i}*,{s}_{j}*){\succ}_{i}f({s}_{i}\prime *,{s}_{j}\prime *)$ and $f({s}_{i}*,{s}_{j}*)$ $~$

_{j}$f({s}_{i}\prime *,{s}_{j}\prime *)$. If the player confirming the agreement is player j, then either $({s}_{i}*,{s}_{j}*)$ or $({s}_{i}\prime *,{s}_{j}\prime *)$ can be confirmed. Hence, both agreements can be confirmed in equilibrium, with player j being indifferent between the two. If the player confirming the agreement is player i, if she is given the possibility to confirm $({s}_{i}*,{s}_{j}*)$, she certainly does it. If she is given the possibility to confirm $({s}_{i}\prime *,{s}_{j}\prime *)$, she does it if, by not confirming, player j would confirm an agreement yielding player i a lower payoff than in $({s}_{i}\prime *,{s}_{j}\prime *)$. All this is shown in next example.

**Example 4: Multiple equilibrium confirmed agreements.**As stated in Proposition 1, a non-generic game may have multiple equilibrium confirmed agreements. Figure 7, with a > b > c > d, provides an example of a non-generic original game G with two equilibrium confirmed agreements in CP(G). G is non-generic since player j gets the same payoff for L and R if player i plays S.

**Pareto efficiency of the equilibrium confirmed agreement.**If an equilibrium exists, we can introduce Proposition 2, concerning the Pareto efficiency of the equilibrium confirmed agreement.

**Proposition 2.**Every equilibrium confirmed agreement in CP(G) is weakly Pareto-efficient.

## 4. Confirmed Agreements in Standard Two-Player Games

**Prisoner’s Dilemma.**The original game G is a standard simultaneous-move Prisoner’s Dilemma. The sets of players’ feasible proposals CP(G) coincide with their sets of actions in the original game: S

_{i}= S

_{j}= {Defect, Cooperate}, henceforth {D, C}. Figure 9, with a > b > c > d, shows the simultaneous-move original game and all the possible agreements of CP(G).8

**Proposition 3.**The subgame perfect equilibrium of CP(G) where G is the Prisoner’s Dilemma is unique, and leads, in period 3, to an equilibrium confirmed agreement where both players cooperate (C, C).

**Hawk-Dove Game.**The original game G is the Hawk-Dove simultaneous-move game (see [9]). The set of players’ feasible proposals, which coincides with the set of players’ strategies in the original game G, is S

_{i}= S

_{j}= {Hawk, Dove}, henceforth {H, D}. Figure 11 shows the simultaneous-move original game and, also, all the possible agreements in CP(G). Parameters are such that a > b > c > d.

**Proposition 4.**The subgame perfect equilibrium of CP(G) where G is the Hawk-Dove Game is unique, and leads, in period 3, to an equilibrium confirmed agreement where both players cooperate (D, D).

**Trust Game.**The original game G is the Trust Minigame, a two-stage game with both the trustor and the trustee having only two possible actions (see [10]). Player i (the trustor) decides whether to Trust (T) or to Not trust (N) player j (the trustee). In case i trusts j, total profits are higher. In that case, j would decide whether to Grab (G) or to Share (S) the higher profits. The strategic form of the Trust Minigame is depicted in Figure 13, where x :=“x if T”, with $x=G,S$, and b

_{i}> c

_{i}> d, a > b

_{j}> c

_{j}, a + d = b

_{i}+ b

_{j}. This figure also represents all the possible agreements of CP(G).

**Proposition 5**. The equilibrium confirmed agreement of CP(G) where G is the Trust Minigame is unique, and leads player i to trust player j and player j to share the higher profits (T,S).

_{j}for player j: when player j has the possibility to get b

_{j}by confirmation, she does it.

_{j}. Moreover, player i would confirm T after history (T,S). Consequently, the equilibrium terminal histories are (N,S,T,S), (T,G,N,S,T,S) and (T,S,T). All lead to the same equilibrium confirmed agreement (T,S).

_{j}. Consequently, the equilibrium terminal histories are (G,N,S,T,S) and (S,T,S). All lead to the same equilibrium confirmed agreement (T,S).

**Ultimatum Game.**The original game G is the Ultimatum Minigame, a two-stage game with both the proposer and the respondent having only two possible actions (see [11]). In the original game G, i (proposer) can offer a fair (F) or unfair (U) division to j (respondent); the latter, after having received i’s offer, may either accept (A) or reject (R). The set of i’s possible strategies coincides with the set of her possible actions, while the set of j’s possible strategies is S

_{j}= {AA,AR,RA,RR}, with xy := “x if F and y if U”, with $x=A,R$ and $y=A,R$. The strategic form of the Ultimatum Minigame in Figure 15 (with a > b > c > d) also represents all the possible agreements of CP(G).9

**Proposition 6.**There are two payoff-equivalent equilibrium confirmed agreements of CP(G) where G is the Ultimatum Minigame, both leading to the egalitarian outcome in G.

## 5. Relevance of Two-Player and Possible Extensions to n-Player Original Game

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Friedman, J.W. A non-cooperative equilibrium for supergames. Rev. Econ. Stud.
**1971**, 38, 1–12. [Google Scholar] [CrossRef] - Smale, S. The prisoner’s dilemma and dynamical systems associated to non-cooperative games. Econometrica
**1980**, 48, 1617–1634. [Google Scholar] [CrossRef] - Cubitt, R.; Sugden, R. Rationally justifiable play and the theory of non-cooperative games. Econ. J.
**1994**, 104, 798–803. [Google Scholar] [CrossRef] - Muthoo, A. Bargaining without commitment. Games Econ. Behav.
**1990**, 2, 291–297. [Google Scholar] [CrossRef] - Rubinstein, A. Perfect equilibrium in a bargaining model. Econometrica
**1982**, 50, 97–109. [Google Scholar] [CrossRef] - Hart, S.; Mas-Colell, A. Bargaining and cooperation in strategic form games. J. Eur. Econ. Assoc.
**2010**, 8, 7–33. [Google Scholar] [CrossRef] - Brams, S.J. Theory of Moves; Cambridge University Press: Cambridge, UK, 1994. [Google Scholar]
- Attanasi, G.; García-Gallego, A.; Georgantzís, N.; Montesano, A. An experiment on prisoner’s dilemma with confirmed proposals. Organ. Behav. Hum. Decis. Process.
**2013**, 120, 216–227. [Google Scholar] [CrossRef] - Smith, J.M.; Price, G.R. The logic of animal conflict. Nature
**1973**, 246, 15–18. [Google Scholar] [CrossRef] - Attanasi, G.; Battigalli, P.; Manzoni, E. Incomplete information models of guilt aversion in the trust game. Manag. Sci.
**2015**, in press. [Google Scholar] - Binmore, K.; Gale, J.; Samuelson, L. Learning to be imperfect: The ultimatum game. Games Econ. Behav.
**1995**, 8, 56–90. [Google Scholar] - Güth, W.; Schmittberger, R.; Schwarze, B. An experimental analysis of ultimatum bargaining. J. Econ. Behav. Organ.
**1982**, 3, 367–388. [Google Scholar] [CrossRef] - Muthoo, A. A note on bargaining over a finite number of feasible agreements. Econ. Theory
**1991**, 1, 290–292. [Google Scholar] [CrossRef] - Sutton, J. Non-cooperative bargaining theory: An introduction. Rev. Econ. Stud.
**1986**, 53, 709–724. [Google Scholar] [CrossRef] - Osborne, M.J.; Rubinstein, A. Bargaining and Markets; Academic Press: San Diego, CA, USA, 1990. [Google Scholar]
- Torstensson, P. An n-person Rubinstein bargaining game. Int. Game Theory Rev.
**2009**, 11, 111–115. [Google Scholar] [CrossRef] - Jun, B.H. A structural consideration on 3-person bargaining. Ph.D. Thesis, University of Pennsylvania, Philadelphia, PA, USA, 1987. [Google Scholar]
- Chae, S.; Yang, J. The unique perfect equilibrium of an n-person bargaining game. Econ. Lett.
**1988**, 28, 221–223. [Google Scholar] [CrossRef] - Chae, S.; Yang, J. An n-person pure bargaining game. J. Econ. Theory
**1994**, 62, 86–102. [Google Scholar] [CrossRef] - Krishna, V.; Serrano, R. Multilateral bargaining. Rev. Econ. Stud.
**1996**, 63, 61–80. [Google Scholar] [CrossRef] - Asheim, G.B. A unique solution to n-person sequential bargaining. Games Econ. Behav.
**1992**, 4, 169–181. [Google Scholar] [CrossRef] - Greenberg, J. The Theory of Social Situations: An Alternative Game-Theoretic Approach; Cambridge University Press: Cambridge, UK, 1990. [Google Scholar]
- Cooper, R.; DeJong, D.V.; Forsythe, R.E.; Ross, T.W. Cooperation without reputation: Experimental evidence from prisoner’s dilemma games. Games Econ. Behav.
**1996**, 12, 187–218. [Google Scholar] [CrossRef] - Neugebauer, T.; Poulsen, A.; Schram, A. Fairness and reciprocity in the hawk-dove game. J. Econ. Behav. Organ.
**2008**, 66, 243–250. [Google Scholar] [CrossRef] - Buskens, V.; Raub, W. Rational choice research on social dilemmas: Embeddedness effects on trust. In Handbook of Rational Choice Social Research; Wittek, R., Snijders, T., Nee, V., Eds.; Russell Sage: New York, NY, USA, 2013; pp. 113–150. [Google Scholar]
- Brandts, J.; Cabrales, A.; Charness, G. Forward induction and entry deterrence: An experiment. Econ. Theory
**2006**, 33, 183–209. [Google Scholar] [CrossRef] - Güth, W.; Huck, S.; Muller, W. The relevance of equal splits in ultimatum games. Games Econ. Behav.
**2001**, 37, 161–169. [Google Scholar] [CrossRef]

^{2}This structure of confirmation—proposing twice consecutively the same strategy means confirming it—can be interpreted as a chain between proposals. Attanasi et al. [8] examine the non-chained case with alternating proposals: the first mover starts proposing her strategy, then the second mover counterproposes her strategy, finally the first mover confirms or not the strategy profile. In the former case, the bargaining process ends and the confirmed strategies are played in the original game. In the latter case, the bargaining process restarts without any constraint due to the proposals made before.^{3}The first mover in CP(G) either can be selected at random or the players should agree over her identity. However, for many original games, the identity of the first mover in CP(G) is irrelevant for the equilibrium confirmed agreement. In particular, this is irrelevant for all original games considered in this paper. An original game where the identity of the first mover is relevant for the equilibrium confirmed agreement obtained in CP(G) is the Battle of Sexes (see footnote 7).^{4}Being S_{k}independent of t, we omit the superscript t when indicating the set of possible proposals in period t.^{5}A possible pair of strategies leading to the unique equilibrium agreement (S, R) in CP(G) of Figure 2a is the following: . This pair of strategies is one of the three subgame perfect equilibria when both players are patient, and the unique subgame perfect equilibrium when they are impatient.^{6}Indeed, the sub-tree in periods 3–7, after the sequence of proposals (S, R), coincides with the sub-tree in periods 7–11 after j’s proposal R in period 6. The same holds for periods 11–15, 15–19, …^{7}An example is given by the Battle of Sexes Game, which is not analyzed here. The equilibrium confirmed agreement for CP(G) when the first mover is player i coincides with the Nash equilibrium for G that is more convenient for player j, and vice versa: in this example a second-mover advantage emerges.^{8}Notice that the Prisoner’s Dilemma has been introduced in the game-theoretical literature by explicitly excluding the possibility of bargaining. Therefore, by allowing the two prisoners to play CP(G) before playing G, we end up examining a different strategic situation. However, experimental studies seem to suggest that players behave as if an implicit bargaining occurs (see footnote 10).^{9}Recall that confirmation is achieved through re-proposal of the same strategy of G. Thus a history like (AR,F,AA) is not a terminal history for CP(G) when j is the first mover, even though both strategy profiles (F,AR) and (F,AA) induce the same terminal history in the original game G.^{10}Torstensson [16] has shown that this is not the case when players demand shares for themselves instead of proposing agreements to each other. However, although it is possible to rule out agreements, the majority remains to be subgame perfect equilibrium outcomes.^{11}Krishna and Serrano [20] have proposed a modification of the Jun-Chae-Yang n-player extension of [5], where offers are made to all the players simultaneously and thus the bargaining is multilateral. If, say, at period 2 one player accepts player one’s proposal and the other n–2 players simultaneously reject it, player two “exits” the game in period 2, with player one representing her in any future negotiations. But player one (having failed to let all players accept her proposal) will not be the next proposer. One of the remaining n–2 players is randomly selected to be the proposer in the next bargaining period. Also this mechanism leads to a unique subgame perfect equilibrium. A possible extension of our CP(G) in this direction would lead to greater implementation problems than those characterizing the two extensions analyzed above.

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**MDPI and ACS Style**

Attanasi, G.; García-Gallego, A.; Georgantzís, N.; Montesano, A.
Bargaining over Strategies of Non-Cooperative Games. *Games* **2015**, *6*, 273-298.
https://doi.org/10.3390/g6030273

**AMA Style**

Attanasi G, García-Gallego A, Georgantzís N, Montesano A.
Bargaining over Strategies of Non-Cooperative Games. *Games*. 2015; 6(3):273-298.
https://doi.org/10.3390/g6030273

**Chicago/Turabian Style**

Attanasi, Giuseppe, Aurora García-Gallego, Nikolaos Georgantzís, and Aldo Montesano.
2015. "Bargaining over Strategies of Non-Cooperative Games" *Games* 6, no. 3: 273-298.
https://doi.org/10.3390/g6030273