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.
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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
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.
References
Bresnahan, T. F. (1981). Duopoly models with consistent conjectures. American Economic Review, 71, 934–945.
Cabral, L. M. B. (1995). Conjectural variations as a reduced form. Economics Letters, 49, 397–402.
Corchón, L. C. (1994). Comparative statics for aggregative games. The strong concavity case. Mathematical Social Sciences, 28, 151–165.
Cornes, R. (2016). Aggregative environmental games. Environmental and Resource Economics, 63, 339–365.
Cornes, R., & Hartley, R. (2003). Risk aversion, heterogeneity and contests. Public Choice, 117, 1–25.
Cornes, R., & Hartley, R. (2005). Asymmetric contests with general technologies. Economic Theory, 26, 923–946.
Cornes, R., & Hartley, R. (2007). Aggregative public good games. Journal of Public Economic Theory, 9, 201–219.
Cornes, R., & Hartley, R. (2011). Well-behaved aggregative games. Presented at the 2011 Workshop on Aggregative Games at Strathclyde University, Glasgow.
Cornes, R., & Hartley, R. (2012). Fully aggregative games. Economics Letters, 116, 631–633.
Cornes, R., & Sandler, T. (1983). On commons and tragedies. American Economic Review, 73, 787–792.
Cornes, R., & Sandler, T. (1984a). The theory of public goods: Non-nash behavior. Journal of Public Economics, 23, 367–379.
Cornes, R., & Sandler, T. (1984b). Easy riders, joint production and public goods. Economic Journal, 94, 580–598.
Cornes, R., & Sandler, T. (1996) The theory of externalities, public goods and club goods (2nd ed.). Cambridge: Cambridge University Press.
Costrell, R. M. (1991). Immiserizing growth with semi-public goods under consistent conjectures. Journal of Public Economics, 45, 383–389.
Dixon, H. D., & Somma, E. (2003). The evolution of consistent conjectures. Journal of Economic Behavior & Organization, 51, 523–536.
Dockner, E. J. (1992). Dynamic theory of conjectural variations. Journal of Industrial Economics, 40, 377–395.
Güth, W., & Yaari, M. (1992). An evolutionary approach to explain reciprocal behavior in a simple strategic game. In U. Witt (Ed.), Explaining process and change – Approaches to evolutionary economics (pp. 23–34). Ann Arbor: University of Michigan Press.
Itaya, J.-I., & Dasgupta, D. (1995). Dynamics, consistent conjectures, and heterogeneous agents in the private provision of public goods. Public Finance/Finances Publiques, 50, 371–389.
Itaya, J.-I., & Okamura, M. (2003). Conjectural variations and voluntary public good provision in a repeated game setting. Journal of Public Economic Theory, 5, 51–66.
Laitner, J. (1980). Rational duopoly equilibria. Quarterly Journal of Economics, 95, 641–662.
Makowski, L. (1987). Are ‘rational conjectures’ rational? Journal of Industrial Economics, 36, 35–47.
Maynard Smith, J., & Price, G. R. (1973). The logic of animal conflict. Nature, 246(Nov 2), 15–18.
Müller, W., & Normann, H.-T. (2005). Conjectural variations and evolutionary stability: A rationale for consistency. Journal of Institutional and Theoretical Economics, 161, 491–502.
Perry, M. K. (1982). Oligopoly and consistent conjectural variations. Bell Journal of Economics, 13, 197–205.
Possajennikov, A. (2009). The evolutionary stability of constant consistent conjectures. Journal of Economic Behavior & Organization, 72, 21–29.
Possajennikov, A. (2015). Conjectural variations in aggregative games: An evolutionary perspective. Mathematical Social Sciences, 77, 55–61.
Selten, R. (1980). A note on evolutionarily stable strategies in asymmetric animal conflicts. Journal of Theoretical Biology, 84, 93–101.
Sugden, R. (1985). Consistent conjectures and voluntary contributions to public goods: Why the conventional theory does not work. Journal of Public Economics, 27, 117–124.
Tullock, G. (1980). Efficient rent-seeking. In J. M. Buchanan, R. D. Tollison, & G. Tullock (Eds.), Toward a theory of the rent-seeking society (pp. 131–146). College Station: Texas A&M University Press.
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.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this chapter
Cite this chapter
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
Download citation
DOI: https://doi.org/10.1007/978-3-319-49442-5_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-49441-8
Online ISBN: 978-3-319-49442-5
eBook Packages: Economics and FinanceEconomics and Finance (R0)