The Public goods game is a standard of experimental economics; in the basic game subjects secretly choose how many of their private tokens to put into the public pot. Each subject keeps the tokens they do not contribute plus an even split of the tokens in the pot (researchers running the game multiply the number of tokens in the pot before it is distributed to encourage contribution).
The group as a whole does best when everyone contributes all of their tokens into the public pool. If everyone puts every token they start with into the pot then the group will extract the maximum total reward from the economists running the test. However, the Nash equilibrium in this game is simply zero contributions by all; if the experiment were a purely analytical exercise in game theory it would resolve to zero contributions because any player does better contributing zero than any other amount regardless of whatever anyone else does. (Following game theory, those who contribute nothing are called "defectors", as opposed to the contributors who are called "cooperators". The defector is also a "free rider").
In fact, the Nash equilibrium is rarely seen in experiments; people do tend to add something into the pot. The actual levels of contribution found in individual subjects varies widely (anywhere from 0% to 100% of initial endowment can be chipped in - subjects are heterogeneous). (This conclusion and others taken from Ahn & Janssen's 2003 summary cited below.)
"Repeat-play" public goods games simply involve the same group of subjects playing the basic game over a series of rounds. The typical result is a declining proportion of public contribution, from the simple game (the "One-shot" public goods game). When trusting contributors see that not everyone is giving up as much as they do they tend to reduce the amount they share with the group if the game is repeated to another round. If this is again repeated the same thing happens but from a lower base, so that the amount contributed to the pot is reduced again. However, the amount contributed to the pool rarely drops to zero when rounds of the game are iterated, because there tend to remain a hardcore of ‘givers’.
One explanation for the dropping level of contribution is inequity aversion; once it is realized that others are receiving a bigger share for a smaller contribution the sharing members react against the perceived injustice (even though the identity of the “free riders” are unknown, and it’s only a game). Those who contribute nothing in one round, rarely contribute something in later rounds, even after discovering that other people are.
If the amount contributed isn't hidden it tends to be higher. In a typical public goods game there might be six subjects contributing to the pot so concealing the level of contribution isn't difficult. In "pairwise iterations" with only two players the other player's contribution level is always known.
Famously, the option to punish non-contributors after a round of the public goods game is widely exercised (although costly and technically “irrational”). In most experiments this leads to greater group cooperation, and fewer defections in subsequent rounds.
At the same time, the option to mercy/reward the contributors after a round of the public goods game is exercised also (although more costly and technically “irrational”). But without the helping of punishment mechanism in experiments, mercy/reward seems not enhance long-term group cooperation(Bin Xu, 2008).
In order for contribution to be privately "irrational" the tokens in the pot must be multiplied by an amount smaller than the number of players and greater than 1. Other than this, the level of multiplication has little bearing on strategy, but higher factors produce higher proportions of contribution.
E.g: Isaac et al. showed that with a large group (40) and very low multiplication factor (1.03) almost no-one contributes anything after a few iterations of the game (a few still do). However, with the same size group and a 1.3 multiplication factor the average level of initial endowment contributed to the pot is around 50%.
The name of the game comes from economist’s definition of a “public good”. One type of public good is a costly, "non-excludable" project that every one can benefit from, regardless of how much they contribute to create it (because no one can be excluded from using it - like street lighting). Part of the economic theory of public goods is that they would be under-provided (at a rate lower than the ‘social optimum’) because individuals had no private motive to contribute (the free rider problem). The “public goods game” is designed to test this belief and connected theories of social behaviour.
The empirical fact that subjects in most societies contribute anything at all in the simple public goods game is a challenge for game theory to explain via a motive of total self-interest, although it can do better with the ‘punishment’ variant, or the ‘iterated’ variant; because some of the motivation to contribute is now purely “rational”, if players assume that others may act irrationally and punish them for non-contribution.