Abstract
We justify risk neutral equilibrium bidding in commonly known fair division games with incomplete information by an evolutionary setup postulating (i) minimal common knowledge, (ii) optimal responses to conjectural beliefs how others behave and (iii) evolutionary selection of conjectural beliefs with fitness measured by expected payoffs. After justifying the game forms we derive the evolutionary games for first- and second-price fair division and determine the evolutionarily stable conjectures. The latter coincide with equilibrium bidding, irrespectively of the number of bidders, i.e., heuristic belief adaptation can imply the same bidding behavior as equilibrium analysis based on common knowledge and counterfactual bidding.
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Notes
See Weibull (1995) for the definition and analysis of evolutionarily stable strategies (ESS) which are essentially refined symmetric equilibria of symmetric games.
This is similar to voting rules letting election outcomes only depend on voting results.
Güth (1986) introduces and applies envyfreeness of net trades with respect to bids which, together with incentive compatibility in the sense of truthful bidding being (weakly) undominated, allows to characterise the game form of the second-price auction. Instead of the latter Güth (2011) additionally requires equal net gains with respect to bids what allows to characterise game forms of various bidding contests relying on the first-price rule.
van Damme (1985) argued that envy-freeness according to bids does not guarantee envy-free net exchanges according to values. But as values are only privately known this could not be checked interpersonally and objectively as, for instance, required for legally implementable rules.
(See also Güth and van Damme 1986, equation III.13) who assume a uniform distribution of values defined on (0, 1) and study \(\lambda \)-pricing rules where \(\lambda =0\) and \(\lambda =1\) represent the first- and second-pricing rules, respectively—for \(\lambda \in [0,1]\), the pricing rule is a convex combination of the highest and second-highest bids.
For example, when F is a Beta distribution \(B(v,\alpha ,1)\), the equilibrium bidding strategies for the first-price rule when \(\alpha =1\) and \(n=4\) are identical to those when \(\alpha =2\) and \(n=2\). This pairwise equivalence holds for the first-price rule for \(n=\{6,8,10,\ldots \}\) with \(\alpha =1\) and for \(n=\{3,4,5,\ldots \}\) with \(\alpha =2\). Similarly, when F is \(B(v,1,\beta )\), the equilibrium strategies for the second-price rule when \(\beta =1\) and \(n=4\) are identical to those when \(\beta =2\) and \(n=2\); and the equivalence holds for \(n=\{6,8,10,\ldots \}\) with \(\beta =1\) and for \(n=\{3,4,5,\ldots \}\) with \(\beta =2\).
We do not address the question of the dynamic stability of evolutionarily stable strategies, when the strategy space is continuous as this depends on the measure of closeness of mutations and is beyond the scope of the paper—See e.g., (Oechssler and Riedel 2002, Example 1) who show that a strict equilibrium can be unstable against the whole population shifting away to q’s that are closeby.
We motivate the use of affine linear bidding functions with intercept terms (c or d) depending on the slope (q or p, respectively) to keep the model tractable when trying to determine which monomorphic population composition is evolutionarily stable.
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We thank an anonymous referee for useful comments. Support from the Max Planck Institute for Research on Collective Goods and the Australian Research Council (DP140102949) is gratefully acknowledged. The usual disclaimer applies.
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Güth, W., Pezanis-Christou, P. An indirect evolutionary justification of risk neutral bidding in fair division games. Int J Game Theory 50, 63–74 (2021). https://doi.org/10.1007/s00182-020-00739-9
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DOI: https://doi.org/10.1007/s00182-020-00739-9