From Jungle to Civilized Economy: The Power Foundation of Exchange Economy Equilibrium

Mordechai E. Schwarz

doi:10.1515/bejte-2017-0085

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From Jungle to Civilized Economy: The Power Foundation of Exchange Economy Equilibrium

Mordechai E. Schwarz

Published/Copyright: February 2, 2019

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From the journal The B.E. Journal of Theoretical Economics Volume 19 Issue 2

Abstract

This article explores the evolution of a civilized exchange economy from an anarchistic environment. I analyze a model of stochastic jungle bargaining mechanism and show that it implements the Talmud Rule allocation (Aumann, R. J., and M. Maschler. 1985. “Game Theoretic Analysis of a Bankruptcy Problem from the Talmud.” Journal of Economic Theory 36 (2): 195–213.) in subgame perfect equilibrium. This Pareto-inefficient allocation constitutes the initial endowment of a stable exchange economy and supports stable Walrasian equilibria, implying that civilized economies could evolve from a Hobbesian state of nature without social contract or regulator. The moral implications of these results are also briefly discussed.

Keywords: bankruptcy; contest; anarchy; jungle

JEL Classification: D63; D71; D74; H0; O10

Acknowledgements

I am deeply indebted to Ariel Rubinstein for his endless patience, priceless help and very helpful comments, and to Ronen Bar-El and an anonymous referee for valuable detailed comments and suggestions that improved my work a lot. Of course, the responsibility for every error is solely mine.

Appendix

Proof of Proposition 1:

By the CARA assumption, di=dj⇒ei∗=ej∗, ∀i,j∈N (Skaperdas and Gan 1995). By the symmetry assumption, ei∗=ej∗⇒pi∗e∗=pj∗e∗=12, ∀i,j∈N. Thus, di=dj⇒Ey∗=ms+12d, ∀i,j∈N, compatible with the Talmud Rule allocation.

To see that di=dj, ∀i,j∈N, recall that di=minci,xi+j−maxxi+j−ci,0, implying the following four contingencies:

(a)ci,cj>xi+j⇒di=dj=xi+j

(b)ci>xi+j, cj<xi+j⇒di=dj=cj

(c)ci<xi+j, cj>xi+j⇒di=dj=cj

(d)ci,cj<xi+j⇒di=dj=ci+cj−xi+j. ■

Proof of Proposition 2:

Consider a bargaining solution yTAL,0 where yTAL=ms+12d. By definition yiTAL+yjTAL=x and msi<yiTAL<ci i∈1,2, implying that yTAL,0∈B. By Proposition 1, p1∗=p2∗=12 implying that y∗,e∗=yTAL,e∗. By our assumptions uiyTAL,0>uiyTAL,e∗ ∀i∈N, implying that yTAL,0∈Σx,c thus Σx,c≠∅. ■

Proof of Proposition 3:

Suppose, without loss of generality, that agent i offers agent j a compromise allocation y,0 such that msj⩽yj<yjTAL. Under common knowledge of rationality, agent j knows that even if biyTAL,0<0, since uiyiTAL,0>uiyi∗,ei∗ rational agent i would not go on a duel even if agent j rejects y,0 and insists on yTAL,0. It follows that under common knowledge of rationality, duel threats in order to impose an offer y,0≠yTAL,0 are non-credible thus yTAL,0 is the unique jungle stable bargaining compromise allocation. ■

Proof of Proposition 4:

By definition msi⩽yiTAL⩽ci ∀i∈N and ∑i∈NyiTAL=∑i∈Nmsi+1nd=x, thus yTAL,0∈B. It is left to show that yTAL,0⇒Ry,0=∅.

Aumann and Maschler (1985) showed that the Talmud Rule is consistent. That is, for any coalition S⊆N the allocation of yS among the members of S is in accordance with the Talmud Rule. In particular, ∀i,j⊆N the division of yi+jTAL between i and j satisfies ylTAL=msl+12dl ∀l∈i,j. By Proposition 3, in this case, no agent can benefit by deviation and challenging another agent with an offer, implying that yTAL,0∈Σn,x,c thus Σn,x,c≠∅. ■

Proof of Proposition 5:

By Proposition 4 yTAL,0∈Σn,x,c≠∅. So it is left to prove the reverse direction, namely Σn,x,c≠∅⇒y,0=yTAL,0.

Assume common knowledge of rationality and let y,0∈Σn,x,c where y,0≠yTAL,0. By assumption, Ry,0=∅. However, by Proposition 3, under common knowledge of rationality y,0≠yTAL,0⇒Ry,0≠∅. Therefore y,0∈Σn,x,c⇔y,0=yTAL,0. ■

Proof of Lemma 1:

The first direction iymaxt∈Ryt,0⇒Ryt,0≠∅ is obvious. For the reverse direction, Ryt,0≠∅⇒iymaxt∈Ryt,0, suppose that Ryt,0≠∅ while iymaxt∉Ryt,0. That is, ∃i∈N incentivized to challenge agent j∈N∖iymaxt with an offer yi+jt+1,0, but on the same time agent i is not incentivized to challenge agent iymaxt. The contradiction stems from the fact that the equilibrium compromise share of agent i increases monotonically with yi+j. Thus, given that iymaxt is the wealthiest agent, agent i cannot benefit from challenging agent j more than from challenging agent iymaxt. It follows that j∈Rjyt,0⇒iymaxt∈Rjyt,0, implying that Ryt,0≠∅⇒iymaxt∈Ryt,0. ■

Proof of Lemma 2:

By Proposition 3, agent i concedes to j,

(5)12di+j=12ci−msi,j+cj−msj,i.

Inserting msi=maxxi+j−cj,0 into (5) yields,

(6)12di+j=12ci−maxxi+j−cj,0+ci−maxxi+j−ci,0,

implying that the compromise that requires minimal concession is with the agent j who has the minimal claim. Now recall that bityt,0=uimsi,0−uiyt,0, by definition. Inserting msi into bityt,0 yields

(7)bityt,0=uimaxxi+j−cj,0,0−uiyt,0,

and it follows that the agent with the minimal claim is i1bt. ■

Proof of Proposition 6:

By Lemma 1, Ryt,0≠∅⇒iymaxt∈Ryt,0, and by Lemma 2, the best response strategy of agent iymaxt is to settle with agent i1bt first, implying that at period t+1 agent i1bt must be better off, hence bt+1≼Lbt. Since the process repeats recursively, at period t+2 agent iymaxt+1 settles first with agent i1bt+1, implying that bt+2≼Lbt+1 and so on. It follows that ∀Ryt,0≠∅ there is a finite T such that bTyT,0≼Lby,0, ∀y,0∈Σn,x,c, implying that yT,0 is the nucleolus of the corresponding cooperative game vx,c by definition. Since the nucleolus of vx,c coincides with the Talmud Rule allocation as mentioned above, yT,0=yTAL,0, and by Proposition 5 yT,0=yTAL,0⇒RyT,0=∅. ■

Proof of Proposition 7:

The existence of a Walrasian equilibrium y,0,q in an exchange economy and its Pareto-efficiency were founded by the Arrow and Debreu (1954) and the first welfare theorems, respectively.^[39] It is left to show that under common knowledge of rationality a Walrasian equilibrium is compatible with subgame-perfectness in the stochastic jungle if and only if it relates to yTAL,0 as the initial endowment.

The irrelevancy of x,0 as an initial endowment in the stochastic jungle stems directly from Proposition 6. Therefore, the real choice facing the jungle society is between two game-forms: the stochastic jungle mechanism or the civilized exchange economy with yTAL,0 as initial endowment and P as the unique exchange set.

Let y,0,q∈FP be a Walrasian equilibrium. By Proposition 5 and Proposition 6, the best an agent i∈N can obtain from deviation and challenging another agent j∈N−i with an offer, is yTAL,0. However, y,0,q∈FP implies that uiyTAL,0⩽uiyi,0,q∀i∈N, thus Ry,0,q=∅. On the other hand, y,0,q∉FP implies that there is at least one agent i∈N incentivized to deviate and challenge another agent j∈N−i with a Pareto-improving offer y,0,q∈FP, thus the best response strategy of agent j is to accept, and it follows that a Walrasian equilibrium y,0,q is compatible with a subgame perfect equilibrium in the stochastic jungle if and only if y,0,q∈FP. ■

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Published Online: 2019-02-02

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Keywords for this article

bankruptcy; contest; anarchy; jungle