## Abstract

In a semi-aggregative representation of a game, the payoff of a player depends on a player’s own strategy and on a personalized aggregate of all players’ strategies. Suppose that each player has a conjecture about the reaction of the personalized aggregate to a change in the player’s own strategy. The players play an equilibrium given their conjectures, and evolution selects conjectures that lead to a higher payoff in such an equilibrium. Considering one player role, I show that for any conjectures of the other players, only conjectures that are consistent can be evolutionarily stable, where consistency means that the conjecture is, to a first approximation, correct at equilibrium. I illustrate this result in public good games and contests.

### Keywords

- Evolutionary Stability
- Public Good Game
- Interior Equilibrium
- Pure Public Good
- Strict Nash Equilibrium

*These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.*

Paper prepared for a volume honoring the memory of Richard Cornes. In his time at the University of Nottingham, Richard was a helpful colleague, ready to give advice in his usual witty and entertaining manner.

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## Notes

- 1.
- 2.
Another interpretation is that players first choose conjectures and then play the game. The search is then for an equilibrium in the game of choosing conjectures. I nevertheless prefer the evolutionary interpretation, which makes it clearer that the process of forming beliefs and choosing strategies occur at different times. This evolutionary interpretation is an example of the “indirect evolution approach” (Güth and Yaari 1992).

- 3.
For example, consider the identity

*u*_{ i }(*x*_{1},*…*,*x*_{ n }) =*x*_{ i }+*u*_{ i }(*x*_{1},*…*,*x*_{ n }) −*x*_{ i }. Let*A*_{ i }=*f*_{ i }(*x*_{1},*…*,*x*_{ n }) =*u*_{ i }(*x*_{1},*…*,*x*_{ n }) −*x*_{ i }and write*u*_{ i }(*x*_{ i },*A*_{ i }) =*x*_{ i }+*A*_{ i }. - 4.
In a homogeneous good market with one price, the aggregate (the price) is the same for all firms, and the game is properly aggregative.

- 5.
- 6.
- 7.
Note that the definition focuses on player

*i*treating the other players conjectures as fixed; Selten (1980) showed that such an approach is appropriate in asymmetric games. - 8.
To save space, arguments of derivatives are omitted. It is understood that they are evaluated at

**r**= (*r*_{ i }^{ ES},**r**_{−i }) and CVE (**x**^{∗}(**r**),**A**^{∗}(**r**)). - 9.
The relationship between consistent conjectures and the conjectures that maximize the indirect payoff function

*u*_{ i }(*x*_{ i }^{∗}(*r*_{ i },*r*_{ j }),*A*_{ i }^{∗}(*r*_{ i },*r*_{ j })) was noted by Itaya and Dasgupta (1995) for a two-player public good game. - 10.
- 11.
The observation that the consistent conjecture is the best response conjecture of one player to any given conjecture of the other player was made in Dixon and Somma (2003) for a linear-quadratic Cournot duopoly game.

- 12.
Note that if

*b*_{ i }= 1 for all*i*, then the public good becomes a pure public good and the same aggregate*G*=*∑*_{ i = 1}^{n}*x*_{ i }can be used for all players. - 13.
The second-order condition \(\alpha _{i}(1 -\alpha _{i})(m_{i} - x_{i})^{\alpha _{i}-2}G_{i}^{-\alpha _{i}-1}(-G_{i}^{2} - 2(m_{i} - x_{i})G_{i}r_{i} - (m_{i} - x_{i})^{2}r_{i}^{2}) < 0\) is satisfied for

*r*_{ i }≥ 0 and all interior*x*_{ i },*G*_{ i }. - 14.
Note that if

*b*= 1, then*r*= 0 is the solution of the consistency condition (Sugden 1985). However, for*r*= 0 the solution of the players’ maximization problem is not interior and the first-order conditions do not characterize it. The propositions do not apply in this case. - 15.
To avoid indeterminacies, let \(u_{i} = \frac{1} {n}\) if

*A*= 0. - 16.
For

*n*= 2, there are non-zero symmetric conjectures that are consistent and evolutionarily stable. The consistency condition for*n*= 2 is \(r_{i} = \frac{c_{i}} {c_{j}}r_{j}\). If*c*_{ i }=*c*_{ j }, then any*r*is consistent. It is shown in Possajennikov (2009) that any 0 <*r*< 2 is evolutionarily stable then.

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## Acknowledgements

I would like to thank Dirk Rübbelke and Wolfgang Buchholz for this opportunity. I would also like to thank Alex Dickson for inviting me to participate in April 2011 in a workshop on aggregative games, which incited me to think about conjectures and aggregative games, and develop the ideas leading to this paper. I also thank Maria Montero for improving the exposition in the paper.

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Possajennikov, A. (2017). Evolution of Consistent Conjectures in Semi-aggregative Representation of Games, with Applications to Public Good Games and Contests. In: Buchholz, W., Rübbelke, D. (eds) The Theory of Externalities and Public Goods. Springer, Cham. https://doi.org/10.1007/978-3-319-49442-5_5

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