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Erschienen in:

21.04.2021

# Privacy, Patience, and Protection

verfasst von: Ronen Gradwohl, Rann Smorodinsky

Erschienen in: Dynamic Games and Applications | Ausgabe 4/2021

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## Abstract

We analyze repeated games in which players have private information about their levels of patience and in which they would like to maintain the privacy of this information vis-à-vis third parties. We show that privacy protection in the form of shielding players’ actions from outside observers is harmful, as it limits and sometimes eliminates the possibility of attaining Pareto-optimal payoffs.

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Fußnoten
1
Such an association is corroborated by surveys of individuals’ privacy concerns, such as Rainie et al. [46] and Madden et al. [36].

2
Loosely, a mechanism satisfies $$\varepsilon$$-DP for some individual i if the distribution over the outcomes of the mechanism with i present is $$\varepsilon$$-close to the distribution with i absent [16]. See Dwork [17] for a survey and Abowd and Schmutte [1] for a discussion of DP from an economic point of view.

3
This is captured by the celebrated folk theorem—see, for example, Theorem 13.17 of Maschler et al. [39].

4
See Sect. 5 for a discussion of this assumption on the extreme nature of patience and impatience.

5
This future game may take the form of a bargaining game, à la Rubinstein [49], in which case they would have an advantage if they were to be perceived as patient. Alternatively, the future game could take the form of the buyer–seller bargaining model of Fudenberg and Tirole [22], where, in equilibrium, impatient buyers obtain the goods at lower prices. Given the ambiguity of the future interaction, the buyers would rather not have their type revealed in the current game.

6
Section 4 is significantly longer and more complex than Sect. 3, as the latter is a possibility result that involves an equilibrium construction, whereas the former is an impossibility result that shows that no equilibrium attains particular Pareto-optimal outcomes.

7
As an example of nonexistence, suppose payoffs consist of a sequence of $$10^k$$ 0’s, then $$10^{k+1}$$ 1’s, then $$10^{k+2}$$ 0’s, and so on. In this case the $$\liminf$$ is 0 whereas the $$\limsup$$ is 1, and so the limit does not exist.

8
If we consider payoffs at a random period, then when there is a finite horizon this would equal the mean payoff. A natural extension of this criterion to the game with an infinite horizon is the limit of the finite horizon means, as we propose.

9
We will assume that the long-sighted player incurs a disutility that depends on the prior and the belief after the repeated interaction. The short-sighted player can be modeled in one of two ways: either he also incurs a disutility that depends on the prior and on the belief after the repeated interaction, or he incurs a disutility that depends on the beliefs before and after each stage. We will adopt the latter for simplicity; using the former, however, would not alter our results.

10
The short-sighted type may incur costs at each stage, and so we do not require his costs to be nonnegative to avoid dynamic inconsistency (see [30], for discussion).

11
The existence of the expectations for the costs is guaranteed by the boundedness of $$c_i$$—see, for example, Theorem 4 of Royden and Fitzpatrick [48].

12
Long-run payoffs in our model are the limit-of-means of stage games, and so on the same order of magnitude, but our results would be unchanged if the long-run payoffs were a large multiplicative factor of the limit-of-means, in which case the privacy cost would be small when compared with the long-run payoffs.

13
These are needed for Lemma 4 and consequently one claim of Theorem 2. For any game, they hold with probability 1 following a perturbation of the utilities.

14
An alternative definition is $$\mathrm {PF}=\{v\in V^*: \not \exists v'\in V^*~\text{ s.t. }~v'> v\}$$ (the difference is in the inequality). If $$V^*$$ is a rectangle, for example, then the question is whether the top and right segments are part of $$\mathrm {PF}$$ or just the top right vertex. The choice of definition does not matter for our results.

15
The symmetry and genericity assumptions in the second bullet are necessary for our proof (in particular, for the proof of Lemma 4), but we do not know if they are necessary for the result to hold.

16
Recall that the limit exists with probability 1 by Definition 1.

17
See the proof of Proposition 5 in Fudenberg et al. [21] for details and definitions of $$I_t$$ and $$m^i$$.

18
Note that the cases are not mutually exclusive, and that the latter two would suffice. We include the first for illustrative purposes and because it involves a shorter revelation phase.

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Titel
Privacy, Patience, and Protection
verfasst von
Rann Smorodinsky
Publikationsdatum
21.04.2021
Verlag
Springer US
Erschienen in
Dynamic Games and Applications / Ausgabe 4/2021
Print ISSN: 2153-0785
Elektronische ISSN: 2153-0793
DOI
https://doi.org/10.1007/s13235-021-00386-z

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