skip to main content


Title: Building rational cooperation on their own: Learning to start small
Abstract

We report experimental results for a twice‐played prisoners' dilemma in which the players can choose the allocation of the stakes across the two periods. Our point of departure is the assumption that some (but not all) people are willing to cooperate, as long as their opponent is sufficiently likely to do so. The presence of such types can be exploited to enhance cooperation by structuring the twice‐played prisoners' dilemma to “start small,” so that the second‐stage stakes are larger (but not too much larger) than the first‐stage stakes. We compare conditions where the allocation of stakes is chosen exogenously to conditions where it is chosen by the players themselves. We show that players gravitate toward the payoff‐maximizing strategy of starting small in a twice‐played prisoners' dilemma. Intriguingly, the salutary payoff effects of doing so are larger than those that arise when the same allocation is exogenously chosen.

 
more » « less
Award ID(s):
1658952
NSF-PAR ID:
10069746
Author(s) / Creator(s):
 ;  ;  
Publisher / Repository:
Wiley-Blackwell
Date Published:
Journal Name:
Journal of Public Economic Theory
Volume:
21
Issue:
5
ISSN:
1097-3923
Page Range / eLocation ID:
p. 812-825
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. We report experimental results for a twice-played prisoners’ dilemma in which the players can choose the allocation of the stakes across the two periods. Our point of departure is the assumption that some (but not all) people are willing to cooperate, as long as their opponent is sufficiently likely to do so. The presence of such types can be exploited to enhance cooperation by structuring the twice-played prisoners’ dilemma to “start small,” so that the second-stage stakes are larger (but not too much larger) than the first-stage stakes. We compare conditions where the allocation of stakes is chosen exogenously to conditions where it is chosen by the players themselves. We show that players gravitate toward the payoff maximizing strategy of starting small in a twice-played prisoners’ dilemma. Intriguingly, the salutary payoff effects of doing so are larger than those that arise when the same allocation is exogenously chosen. 
    more » « less
  2. In the past few decades, numerous experiments have shown that humans do not always behave so as to maximize their material payoff. Cooperative behavior when noncooperation is a dominant strategy (with respect to the material payoffs) is particularly puzzling. Here we propose a novel approach to explain cooperation, assuming what Halpern and Pass call translucent players. Typically, players are assumed to be opaque, in the sense that a deviation by one player in a normal-form game does not affect the strategies used by other players. However, a player may believe that if he switches from one strategy to another, the fact that he chooses to switch may be visible to the other players. For example, if he chooses to defect in Prisoner’s Dilemma, the other player may sense his guilt. We show that by assuming translucent players, we can recover many of the regularities observed in human behavior in well-studied games such as Prisoner’s Dilemma, Traveler’s Dilemma, Bertrand Competition, and the Public Goods game. The approach can also be extended to take into account a player’s concerns that his social group (or God) may observe his actions. This extension helps explain prosocial behavior in situations in which previous models of social behavior fail to make correct predictions (e.g. conflict situations and situations where there is a trade-off between equity and efficiency). 
    more » « less
  3. Abstract

    This study examines how exploiting biases in probability judgment can enhance deterrence using a fixed allocation of defensive resources. We investigate attacker anchoring heuristics for conjunctive events with missing information to distort attacker estimates of success for targets with equal defensive resources. We designed and conducted a behavioral experiment functioning as an analog cyber attack with multiple targets requiring three stages of attack to successfully acquire a target. Each stage is associated with a probability of successfully attacking a layer of defense, reflecting the allocation of resources for each layer. There are four types of targets that have nearly equal likelihood of being successfully attacked, including one type with equally distributed success probabilities over every layer and three types with success probabilities that are concentrated to be lowest in the first, second, or third layer. Players are incentivized by a payoff system that offers a reward for successfully attacked targets and a penalty for failed attacks. We collected data from a total of 1,600 separate target selections from 80 players and discovered that the target type with the lowest probability of success on the first layer was least preferred among attackers, providing the greatest deterrent. Targets with equally distributed success probabilities across layers were the next least preferred among attackers, indicating greater deterrence for uniform‐layered defenses compared to defenses that are concentrated at the inner (second or third) levels. This finding is consistent with both attacker anchoring and ambiguity biases and an interpretation of failed attacks as near misses.

     
    more » « less
  4. Mixed strategies are often evaluated based on the expected payoff that they guarantee. This is not always desirable. In this paper, we consider games for which maximizing the expected payoff deviates from the actual goal of the players. To address this issue, we introduce the notion of a (u,p)-maxmin strategy which ensures receiving a minimum utility of u with probability at least p. We then give approximation algorithms for the problem of finding a (u, p)-maxmin strategy for these games. The first game that we consider is Colonel Blotto, a well-studied game that was introduced in 1921. In the Colonel Blotto game, two colonels divide their troops among a set of battlefields. Each battlefield is won by the colonel that puts more troops in it. The payoff of each colonel is the weighted number of battlefields that she wins. We show that maximizing the expected payoff of a player does not necessarily maximize her winning probability for certain applications of Colonel Blotto. For example, in presidential elections, the players’ goal is to maximize the probability of winning more than half of the votes, rather than maximizing the expected number of votes that they get. We give an exact algorithm for a natural variant of continuous version of this game. More generally, we provide constant and logarithmic approximation algorithms for finding (u, p)-maxmin strategies. We also introduce a security game version of Colonel Blotto which we call auditing game. It is played between two players, a defender and an attacker. The goal of the defender is to prevent the attacker from changing the outcome of an instance of Colonel Blotto. Again, maximizing the expected payoff of the defender is not necessarily optimal. Therefore we give a constant approximation for (u, p)-maxmin strategies. 
    more » « less
  5. Abstract

    We present the formulation and optimization of a Runge–Kutta-type time-stepping scheme for solving the shallow-water equations, aimed at substantially increasing the effective allowable time step over that of comparable methods. This scheme, called FB-RK(3,2), uses weighted forward–backward averaging of thickness data to advance the momentum equation. The weights for this averaging are chosen with an optimization process that employs a von Neumann–type analysis, ensuring that the weights maximize the admittable Courant number. Through a simplified local truncation error analysis and numerical experiments, we show that the method is at least second-order in time for any choice of weights and exhibits low dispersion and dissipation errors for well-resolved waves. Further, we show that an optimized FB-RK(3,2) can take time steps up to 2.8 times as large as a popular three-stage, third-order strong stability-preserving Runge–Kutta method in a quasi-linear test case. In fully nonlinear shallow-water test cases relevant to oceanic and atmospheric flows, FB-RK(3,2) outperforms SSPRK3 in admittable time step by factors roughly between 1.6 and 2.2, making the scheme approximately twice as computationally efficient with little to no effect on solution quality.

    Significance Statement

    The purpose of this work is to develop and optimize time-stepping schemes for models relevant to oceanic and atmospheric flows. Specifically, for the shallow-water equations we optimize for schemes that can take time steps as large as possible while retaining solution quality. We find that our optimized schemes can take time steps between 1.6 and 2.2 times larger than schemes that cost the same number of floating point operations, translating directly to a corresponding speedup. Our ultimate goal is to use these schemes in climate-scale simulations.

     
    more » « less