Source-linked AI summary
A Model of Human Cooperation in Social Dilemmas
Valerio Capraro
TL;DR
The paper addresses how to predict population-average cooperation in social dilemmas and account for observed behavioral patterns. It models players as forecasting coalition play and choosing according to their most optimistic forecast. The authors report satisfactorily accurate predictions and organize several payoff-dependent patterns of cooperation.
Problem
The paper addresses how to predict population-average cooperation in social dilemmas while accounting for observed patterns across games and payoffs.
Source: §4 · p. 10Method
The model compares incentive and risk to estimate abandonment probabilities, converts them into common beliefs, and restricts the original game accordingly.
Source: §4 · p. 9Results
The model makes satisfactorily accurate predictions of population-average behavior and organizes several patterns of cooperation across social dilemmas.
Source: §4 · p. 10Takeaways & Limitations
The model accounts for payoff-related cooperation patterns, including higher cooperation as Traveler’s dilemma penalties decrease and as Public Goods marginal returns increase.
Source: §4 · p. 10Direct comparison between iterated and one-shot games is not possible because iterated games have a richer strategy set, and the model’s formalization merits further investigation and improvement.
Source: §4 · pp. 9–10
Abstract
from arXiv · showhide
Social dilemmas are situations in which collective interests are at odds with private interests: pollution, depletion of natural resources, and intergroup conflicts, are at their core social dilemmas. Because of their multidisciplinarity and their importance, social dilemmas have been studied by economists, biologists, psychologists, sociologists, and political scientists. These studies typically explain tendency to cooperation by dividing people in proself and prosocial types, or appealing to forms of external control or, in iterated social dilemmas, to long-term strategies. But recent experiments have shown that cooperation is possible even in one-shot social dilemmas without forms of external control and the rate of cooperation typically depends on the payoffs. This makes impossible a predictive division between proself and prosocial people and proves that people have attitude to cooperation by nature. The key innovation of this article is in fact to postulate that humans have attitude to cooperation by nature and consequently they do not act a priori as single agents, as assumed by standard economic models, but they forecast how a social dilemma would evolve if they formed coalitions and then they act according to their most optimistic forecast. Formalizing this idea we propose the first predictive model of human cooperation able to organize a number of different experimental findings that are not explained by the standard model. We show also that the model makes satisfactorily accurate quantitative predictions of population average behavior in one-shot social dilemmas.
1. An informal description of the model
The model compares selfish and cooperative coalition forecasts, then restricts play to strategy profiles meeting the best forecast. In a Prisoner’s dilemma example, this yields a cooperative equilibrium halfway between cooperation and defection for both players.
1. An informal description of the model
- The model estimates a player's probability of abandoning cooperation by comparing the incentive to defect with the risk of deviating.In the example, the incentive is 0.05 and the risk is 0.10; the model defines the abandonment probability as incentive divided by incentive plus risk.Source: §1 · p. 4
- The selfish coalition forecasts 0.05, while the cooperative coalition forecasts 0.10, making cooperation the more optimistic forecast.The cooperative coalition’s maximum payoff is 0.15 at (C, C), but the model accounts for possible deviation from that profile.Source: §1 · p. 4
- The best forecast becomes a common payoff threshold: only strategy profiles giving each player at least 0.10 are allowed in the restricted game.Figure 1 depicts the allowed profiles as the red set within the full unit square of mixed strategies.Source: §1 · p. 4, Figure 1 · p. 5
- λ=1/2 is the cooperative equilibrium for both players; reported 58% and 65% cooperation in two treatments.The model selects the restricted-game equilibrium by finding the point in the allowed set closest to (D, D).Source: §1 · p. 4
2. The model
The model assigns payoff forecasts to coalition structures by weighing outcomes under adherence and abandonment. It selects the most optimistic forecast to restrict the game, then defines cooperative equilibrium as a Nash equilibrium of that induced game.
2. The model
- A coalition structure partitions players into coalitions, whose members act as one player in the associated game with summed payoffs.The model starts with symmetric games and two players for simplicity, then describes an extension to N players.Source: §2 · pp. 5–6
- For each coalition structure, the model compares incentives to abandon it with the risks of deviation and uses their ratio to assign an abandonment probability.The authors describe this incentive-risk comparison as intuitive, while noting that other comparisons merit future investigation.Source: §2 · p. 6
- Each player's coalition value is an average of payoff infima across scenarios in which no players or subsets of players abandon the coalition.The weights are prior probabilities for those abandonment scenarios; under symmetry, all players have the same coalition value.Source: §2 · pp. 5–6
- The model chooses a coalition structure with the highest value and restricts the induced game to profiles giving every player at least that value; cooperative equilibrium is a Nash equilibrium of this game.The induced game is independent of which maximizing coalition structure is chosen when several tie.Source: §2 · pp. 6–7
- The model assumes common knowledge that both players use the same coalition-based reasoning; for N players, independence can simplify joint-deviation probabilities, while influence between players remains open to exploration.For the symmetric classical social dilemmas considered, only the fully selfish and fully cooperative coalition structures need to be evaluated.Source: §2 · p. 7
3. Predictions of the model
The cooperative equilibrium yields payoff-sensitive predictions across several social dilemmas. The predictions include increasing cooperation as the Prisoner’s dilemma parameter rises, alongside comparisons with reported experimental behavior.
3. Predictions of the model
- 37% cooperated in the laboratory and 47% on MTurk; the difference was reported as not statistically significant.The comparison concerns a Prisoner’s dilemma with T=10, R=7, P=3, and S=0.Source: §3 · p. 7
- As k rises above 1, the Prisoner’s dilemma prediction moves continuously and monotonically from defection toward cooperation, tending to cooperation as k grows.For k ≤1, the cooperative equilibrium coincides with Nash equilibrium.Source: §3 · p. 8
- At b=5 and b=180, Traveler’s dilemma predictions align with reported high claims and Nash-equilibrium choices; at b=2, the prediction is close to strategy 99.The model predicts smaller claims as the bonus/penalty b increases.Source: §3 · p. 8
- 0.66–0.67 is the model’s cooperative equilibrium at α=0.8, versus a reported mean contribution of 0.50 and modes of 0.60 and 0.80.The reported modes occurred 6 and 5 times out of 32, respectively.Source: §3 · pp. 8–9
- In the Tragedy of the Commons, v(pc) exceeds v(ps) if and only if α > 2; the passage says it and the Public Goods game represent the same strategic situation.Source: §3 · p. 9
- Despite similar average payoffs across outcomes, the cooperative equilibrium predicts significantly higher cooperation in Chicken than in the Prisoner’s dilemma.The comparison draws on iterated-game observations, while the proposed explanation uses the cooperative equilibrium in one-shot games.Source: §3 · p. 9
4. Conclusions
The authors model cooperation by having players forecast coalition outcomes and follow their most optimistic forecast. The model predicts population-average behavior and organizes several payoff-dependent cooperation findings, while individual-level prediction remains a future aim.
4. Conclusions
- The authors argue that cooperation may be intrinsic: players forecast coalition outcomes rather than acting a priori as isolated agents.They link this assumption to experiments finding cooperation in one-shot dilemmas without external controls and payoff-dependent cooperation rates.Source: §4 · p. 9
- Players assess the probability that others abandon collective interests by comparing deviation incentives and risks, producing common beliefs that restrict the original game.The authors describe this formalization as mathematically simple and seemingly natural.Source: §4 · p. 9
- The paper describes Chicken as similar to the Prisoner’s dilemma, but with payoffs ordered T > R > S > P.The footnote also notes two pure Nash equilibria and a payoff-dependent symmetric evolutionarily stable mixed Nash equilibrium.Source: §4 · p. 9
- Iterated and one-shot games cannot be directly compared because iterated games offer a much richer set of strategies.The authors note that behavior differences observed in iterated treatments are predicted for one-shot treatments, but say this connection merits further investigation.Source: §4 · p. 9
- The authors say the model’s formalization is natural but deserves further investigation and possible improvement.They hope to incorporate factors such as family history, age, culture, incentives, and iterations to support individual-level predictions.Source: §4 · p. 10
- The model makes satisfactorily accurate predictions of population-average behavior and organizes payoff-linked cooperation patterns across the Prisoner’s dilemma, Traveler’s dilemma, Public Goods, and Chicken games.Cooperation increases as the cost-benefit ratio, bonus/penalty, or per-capita marginal return decreases, and is higher in Chicken than in the Prisoner’s dilemma with similar payoffs.Source: §4 · p. 10