Abstract
In this chapter I study optimal contracts in the context of probabilistic choice models. Probabilistic choice models represent a more drastic deviation from the standard paradigm since agents in these type of models no longer follow maximizing behavior.
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Notes
- 1.
Situation is more complicated in the multidimensional models. The concept of top type is not well-defined for such models. If the number of quality dimensions equals the dimensionality of type one can prove under some technical conditions that there are not distortions on the boundary of type set in the direction normal to the boundary (Basov 2005b).
- 2.
For a textbook exposition of the Revelation Principle, see Mas-Colell et al. (1995).
- 3.
Recall that the major trade-off is between inefficient provision of the low type and information rents paid to the high type. If the high type is sufficiently numerous it is advantageous to exclude low type completely to avoid paying information rents to the high type.
- 4.
The Luce’s probabilities are also known in Industrial Organization (IO) as the logit probabilities. However, while in the IO literature the justification for probabilistic choice comes from horizontal heterogeneity in the consumers’ tastes, Luce justified it appealing to the bounded rationality of the decision makers.
- 5.
Recall that the binding constraints are the individual rationality for the low type and the incentive compatibility for the high type.
- 6.
Since ψ(·) is strictly increasing, the inverse function \(\psi^{ - 1 } ( \cdot )\) exists.
- 7.
In APET A stands for actions activating agents, P for pattern match, E for emotion, and T for thought. For an exposition of the model, see Griffin and Tyrrell (2003).
- 8.
At t = 0 differentiability is understood as differentiability from the right.
- 9.
This assumption can be interpreted as saying that the environment is sufficiently safe.
- 10.
This logic is a special case of application of the concept of quantal response equilibrium, developed by McKelvey and Palfrey (1996).
- 11.
Generalization to measurable subsets of arbitrary measure space proceeds in a similar way.
- 12.
See AGH for a proof that at any equilibrium bidders with identical valuation choose their bids from the same distribution.
- 13.
Just recall that \(F^{'} \left( b \right) = f\left( b \right).\) Then (5.11) is a first-order differential equation for cumulative distribution function. General solution will depend on arbitrary constant, K, the value of which is determined from the normalization condition.
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Basov, S. (2016). Probabilistic Choice and Optimal Contracts. In: Social Norms, Bounded Rationality and Optimal Contracts. Studies in Economic Theory, vol 30. Springer, Singapore. https://doi.org/10.1007/978-981-10-1041-5_5
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