## Abstract

We consider the simplest version of a jungle economy à la Piccione-Rubinstein, wherein as many agents as goods are assumed, agents consume at most one indivisible good, and a transitive strong power relation exists. We first study the wilderness of jungle equilibria, i.e., whether they are Pareto-minimal (an allocation is Pareto-minimal if it is impossible to reduce the welfare of one agent without increasing the welfare of another). We show that jungle equilibria are not necessarily Pareto-minimal. We then study and characterize the set of Pareto-minimal jungle equilibria. Second, we tackle the case of equally powerful people, in contrast to the assumption that the power relation is asymetric. Assuming specifically a transitive weak power relation, we show that jungle equilibria exist, but that they are not always unique, nor Pareto-optimal. We also provide conditions under which those equilibria are Pareto-minimal.

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## Notes

- 1.
- 2.
- 3.
Pursuing the line of research opened by Piccione and Rubinstein, Houba et al. (2017) study general conditions under which jungle equilibria coincide with lexicographic welfare-maximizing allocations (in which all of the economy’s resources initially are common goods and stronger agents take from the pile of common goods before weaker agents can).

- 4.
The setting is identical to the model used by Feldman and Serrano (2006, chapter 4) or Rubinstein (2012), which is a special case of the general framework analyzed by Piccione and Rubinstein (2007). To avoid confusion, we shall hereafter refer to the first model rather than a general jungle economy.

- 5.
The assumptions mean that it is always possible to compare the powers of two agents, that no two agents are equally powerful, and that if an agent

*i*is more powerful than an agent*j*, and that agent*j*is more powerful than agent*k*, then agent*i*is also more powerful than agent*k*. - 6.
The indivisible good commonly referred to is a house, which is reminiscent of the Shapley and Scarf (1974) paper.

- 7.
That assumption certainly is disputable (but see section 2 of Piccione and Rubinstein 2007). It is, however, instrumental in simplifying the analysis.

- 8.
A function \(f : {{\mathcal {C}}} \rightarrow {\mathbb {R}}\) is injective if it never maps two different elements of \({{\mathcal {C}}}\) to the same element of \({\mathbb {R}}\).

- 9.
The assumption is made in section 4.3 of Piccione and Rubinstein (2007).

- 10.
Notice that the definition rules out free disposal. In particular, the goods cannot be destroyed. Moreover, it is not possible to prevent one or several agents from consuming the goods. I thank a referee for drawing my attention to this point. Clearly, the jungle is not totally wild.

- 11.
Notice that when the utility functions are injective the condition \(u_i\big (x(i)\big ) \ge u_i\big (x(j)\big )\) actually implies \(u_i\big (x(i)\big ) > u_i\big (x(j)\big )\).

- 12.
On the notion of Pareto-minimal point, see, e.g., Luptácǐk (2009, p. 226). As observed by a referee, Pareto-minimality is connected to the notion of dual Pareto efficiency introduced by Luenberger (1994). Specifically, a feasible assignment \((p^*, M^*)\) (

*i.e.*, a list of prices and a distribution of incomes that can be obtained by the workings of a market economy) is dual Pareto-efficient if no other feasible assignment (*p*,*M*) such that \(v_i(p, M_i) \le v_i(p^*, M^*_i)\) is possible for all consumers in the economy and \(v_j(p, M_j) < v_j(p^*, M^*_j)\) for at least one individual. The definition refers to incomes and prices, whereas Pareto-minimality refers to quantities. - 13.
As suggested by a referee, if all the goods could be destroyed, then the only Pareto-minimal allocation would be the nill allocation.

- 14.
In Table 1 column

*i*describes the ranking of agent*i*. - 15.
The meaning of \(j \not \gtrsim i\) is that it is false that

*j*is at least as powerful as*i*. - 16.
I thank a referee for suggesting studying this example.

- 17.
In particular, when all agents are equally powerful any allocation is a jungle equilibrium. That is, however, an extreme case, which can hardly correspond to what is generally referred to as a jungle.

- 18.
That result can be seen by using the allocation studied in the proof of Proposition 4. Since each agent receives the good that he ranks the highest in the set of goods left to him by the agents at least as powerful as him, one cannot devise a Pareto-improving allocation.

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## Acknowledgements

I thank Susan Crettez, Naila Hayek and Merhdad Vahabi, the guest editor, for very helpful comments on a previous version of this work. I also am grateful to three referees for stimulating and constructive remarks on the different versions of the manuscript submitted to the review. Finally, I warmly thank William Shughart, Editor in Chief, for countless suggestions to polish the paper.

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Crettez, B. Pareto-minimality in the jungle.
*Public Choice* **182, **495–508 (2020). https://doi.org/10.1007/s11127-019-00677-5

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### Keywords

- Involuntary exchange
- Jungle economy
- Jungle equilibria
- Pareto-minimal
- Power relation

### JEL Classification

- C7
- D61
- D74
- P48
- P52