Public goods games and conditional cooperation, explained

A public goods game is an experiment in which each member of a group decides how much of a private endowment to put into a shared account that pays every member equally. In the standard linear version, each token put in returns less to the giver than it costs but more to the group, so a selfish player gives nothing and a group of givers does best. John Ledyard's survey of the experiments, published in the 1995 Handbook of Experimental Economics, lists the stylized facts: people start by giving about halfway between nothing and everything, and giving declines with repetition. Later Zurich experiments showed that punishment can sustain giving and that many people cooperate only conditionally.

How the linear game works

The Zurich design shows the arithmetic. In the July 2000 working paper by Fischbacher, Gächter and Fehr, each of four players splits 20 tokens between a private account and a project. A player's payoff is the tokens kept plus 0.4 times the sum of all four contributions. A token kept is worth 1 to its owner. A token contributed returns 0.4 to the giver and 1.6 to the group as a whole.

Ledyard credits Isaac, Walker and Thomas (1984) with the name for that 0.4: the marginal per capita return (MPCR). When the MPCR is below 1, giving nothing is the dominant strategy. Fehr and Gächter note that total group payoff is still largest when everyone gives everything.

Giving starts near half and decays

Ledyard adds a third fact, that face-to-face communication raises giving, and warns that all three deserve skepticism. Ledyard reports that in the Isaac, Walker and Thomas experiments, groups of 4 and 10 played 10 periods known in advance, with an MPCR of 0.3 or 0.75. Average giving was 42 percent of the endowment across treatments and 51 percent in first periods, ranging from 0 to 83 percent. Raising the MPCR from 0.3 to 0.75 raised giving in all cases.

Fehr and Gächter cite a meta-study by Fehr and Schmidt of 12 public goods experiments without punishment. In first periods, average and median giving ran between 40 and 60 percent of the endowment. In the final period, 73 percent of 1,042 subjects gave nothing.

Punishment changes the outcome

Fehr and Gächter's June 1999 working paper, published in the American Economic Review in 2000, added a second stage. After seeing each member's contribution, players could assign punishment points. Each point cut the target's first-stage payoff by 10 percent and cost the punisher. Groups of four had 20 tokens each and an MPCR of 0.4, and played ten periods with and without punishment. In Partner sessions the group stayed fixed. In Stranger sessions it was reshuffled every period.

A selfish player should neither punish nor give. Yet free riders were heavily punished, and the further they fell below the group's standard, the more they were punished. Without punishment, 53 to 75 percent of subjects gave nothing in the final period. With punishment, average giving of 50 to 95 percent of the endowment could be maintained. In the final period of the Partner treatment with punishment, 82.5 percent of subjects gave their whole endowment.

Conditional cooperators

Fischbacher, Gächter and Fehr asked why giving decays. In a one-shot game with no repetition, 44 Zurich undergraduates in 11 groups filled out a contribution table: how much each would give for each of the 21 possible average contributions of the others. They earned 27.6 Swiss francs on average, about $21.

Half the subjects, 22 of 44, were conditional cooperators whose giving rose with others' giving. Four matched others exactly. Only 11.9 percent of their entries were above the matching line, which the authors call conditional cooperation with a self-serving bias. Thirteen subjects, about 30 percent, gave nothing at every level; the published abstract in Economics Letters, June 2001, rounds this to about a third. Six subjects, 14 percent, were hump-shaped, matching others up to 10 tokens and then giving less.

The authors' explanation for decay follows. If some free ride and cooperators give a little less than others, repeated giving spirals downward even when most people are willing to cooperate.

Language-model agents in public goods games

Guzman Piedrahita and colleagues, at the 2025 Conference on Language Modeling (COLM), ran seven agents for 15 rounds with 20 tokens each, a multiplier of 1.6, and a choice each round of joining a group with or without sanctions. Average contributions were 5.39 tokens for o1-mini, 13.71 for GPT-4o, and 18.71 for the best traditional model, against 18.3 for humans in the 2006 experiment by Gürerk and colleagues that the design adapts. People punished more than they rewarded, at a ratio of 1.66, while every model preferred rewards, at ratios of 0.00 to 0.88. Two configurations ran only once because of budget limits.

An August 2026 preprint by Raad Bin Tareaf ran twelve open-weight models in a setting like the Zurich one: four agents, 20 points, ten rounds, and an MPCR of 0.4. First-round giving fell inside the preset human margin for 8 of 11 models, and final-round giving for none. Seven of the eleven ended giving more than they started, and only one declined. A coded conditional cooperator fell from 0.48 to 0.07 of the endowment, as people do. Adding punishment lowered final-round giving in four of nine models. Bin Tareaf notes that the analysis was not checked by an independent analyst.

What to check

• What were the MPCR and group size, given that raising the MPCR from 0.3 to 0.75 raised human giving in every case?
• Does the agent's giving decline over rounds, or does it match only the first round?
• Was a conditional schedule elicited, as in the Zurich table of 21 possible averages?
• Does punishment raise giving, as in Fehr and Gächter, or lower it, as in four of nine models in 2026?
• How many runs per model, and are the human figures from the same design?

Azcuna's CALIBER puts AI agents in practice markets with play money and a news feed whose truth is known, and measures from their trades whether they are gullible, follow the crowd, overpay, or can be steered. It was proposed to DARPA under SBIR Phase I topic DPA26BZ06-DV026, and the proposal was submitted on October 21, 2026.

Sources

1. 1Public Goods: A Survey of Experimental Research · John O. Ledyard, Caltech working paper 861, revised February 1994; published in the Handbook of Experimental Economics, 1995 (CaltechAUTHORS)
2. 2Cooperation and Punishment in Public Goods Experiments · Fehr and Gächter, University of Zurich IEW Working Paper No. 10, June 1999; published in American Economic Review 90(4), 2000
3. 3Are People Conditionally Cooperative? Evidence from a Public Goods Experiment · Fischbacher, Gächter and Fehr, University of Zurich IEW Working Paper No. 16, July 2000
4. 4Are people conditionally cooperative? Evidence from a public goods experiment · Fischbacher, Gächter and Fehr, Economics Letters 71(3), June 2001 (RePEc record and abstract)
5. 5Corrupted by Reasoning: Reasoning Language Models Become Free-Riders in Public Goods Games · arXiv (Guzman Piedrahita, Yang, Sachan, Ramponi, Schölkopf and Jin), version 2, July 2025; COLM 2025
6. 6Benchmarking large language model agent societies against human behavioural distributions · arXiv (Raad Bin Tareaf), version 1, August 2026

Where this shows up in our work

• CALIBER, the market testbed for AI agents measures from an AI agent's trades whether it is gullible, herds, overpays, or can be steered.
• LEDGER, auditable Army acquisition decisions turns a trade study into a package where every number cites its source and AI drafts but never decides.