# Public goods game

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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 a public pot. The tokens in this pot are multiplied by a factor (greater than one and less than the number of players, N) and this "public good" payoff is evenly divided among players. Each subject also keeps the tokens they do not contribute.

## Results

The group's total payoff is maximized when everyone contributes all of their tokens to the public pool. 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 rational agent does best contributing zero, regardless of whatever anyone else does. This only holds if the multiplication factor is less than the number of players, otherwise the Nash equilibrium is for all players to contribute all of their tokens to the public pool. [1]

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 varies widely (anywhere from 0% to 100% of initial endowment can be chipped in). [2] The average contribution typically depends on the multiplication factor. [3] Capraro has proposed a new solution concept for social dilemmas, based on the idea that players forecast if it is worth to act cooperatively and then they act cooperatively in a rate depending on the forecast. His model indeed predicts increasing level of cooperation as the multiplication factor increases. [4]

Depending on the experiment's design, those who contribute below average or nothing are called "defectors" or "free riders", as opposed to the contributors or above average contributors who are called "cooperators". [1]

## Variants

### Iterated public goods games

"Repeat-play" public goods games 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 in the next round. [5] [6] 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 hard core of ‘givers’.

One explanation for the dropping level of contribution is inequity aversion. During repeated games players learn their co-player's inequality aversion in previous rounds on which future beliefs can be based. If players receive 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). [7] Those who contribute nothing in one round, rarely contribute something in later rounds, even after discovering that others are.

### Open public goods games (Transparency)

Transparency about past choices and payoffs of group members affects future choices.  Studies show individuals in groups can be influenced by the group leaders, whether formal or informal, to conform or defect. Players signal their intentions through transparency which allows “conditional operators” to follow the lead. [8] If players are informed of individual payoffs of each member of the group it can lead to a dynamic of players adopting the strategy of the player who benefited the most (contributed the least) in the group. This can lead a drop in cooperation through subsequent iterations of the game. [9] However, if the amount contributed by each group member is not hidden, the amount contributed tends to be significantly higher. [10] The finding is robust in different experiment designs: Whether in "pairwise iterations" with only two players (the other player's contribution level is always known) or in nominations after the end of the experiment.

### Public goods games with punishment and/or reward

The option to punish non-contributors and to reward the highest contributions after a round of the public goods game has been the issue of many experiments. Findings strongly suggest that non-rewarding is not seen as sanction, while rewards don't substitute punishment. Rather they are used completely differently as a means to enforce cooperation and higher payoffs.

Punishing is exercised, even at a cost, and in most experiments it leads to greater group cooperation. [11] However, since punishment is costly, it tends to lead to (marginally) lower payoffs, at least initially. [12] On the other hand, a 2007 study found that rewards alone could not sustain long-term cooperation. [13]

Many studies therefore emphasize the combination of (the threat of) punishment and rewards. The combination seems to yield both a higher level of cooperation and of payoffs. This holds for iterated games in changing groups [11] [13] as well as in identical groups. [12]

### Asymmetric costs and/or benefits

Asymmetric cost and or benefit functions have direct influence in the contribution behaviour of agents. When confronted with different payoff returns to their contributions, agents behave differently though they still contribute more than in Nash equilibrium. [6]

### Income variation

A public goods games variant suggested as an improvement for researching the free rider problem is one in which endowment are earned as income. The standard game (with a fixed initial endowments) allows no work effort variation and cannot capture the marginal substitutions among three factors: private goods, public goods, and leisure. [14]

Researchers have found that in an experiment where an agent's wealth at the end of period t serves as her endowment in t+1, the amounts contributed increase over time even in the absences of punishment strategies. [15]

### Framing

A different framing of the original neutral experiment setting induces players to act differently because they associate different real-life situations. For example, a public good experiment could be presented as a climate negotiation or as contributions to private parties.

The effect of associations (label frame) depends on the experience-pool the player made with similar real-life frames. This is especially true for one-shot (not iterated) games where players can only infer others’ behavior and expectations from their life experiences. Therefore, the same frame can induce more and also less contribution, even in similar cultures. Label frames move beliefs i.e. about other player's behaviour, and these beliefs subsequently shape motivation and choice. [16]

Also, the same game structure can always be presented as a gain or a loss game. Because of the Framing effect players respond completely differently when it is presented as a gain or a loss. If public good games are presented as a loss, i.e. a player's contribution in a private engagement diminishes other player's payoff, contributions are significantly lower. [17]

## Multiplication factor

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.

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%. [18]

## Implications

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.

### Game theory

The empirical fact that subjects in most societies contribute anything 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.

### Applications to sociology

The sociological interpretation of these results emphasizes group cohesion and cultural norms to explain the "prosocial" outcomes of public goods games.

## Related Research Articles

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The prisoner's dilemma is a standard example of a game analyzed in game theory that shows why two completely rational individuals might not cooperate, even if it appears that it is in their best interests to do so. It was originally framed by Merrill Flood and Melvin Dresher while working at RAND in 1950. Albert W. Tucker formalized the game with prison sentence rewards and named it "prisoner's dilemma", presenting it as follows:

Two members of a criminal gang are arrested and imprisoned. Each prisoner is in solitary confinement with no means of communicating with the other. The prosecutors lack sufficient evidence to convict the pair on the principal charge, but they have enough to convict both on a lesser charge. Simultaneously, the prosecutors offer each prisoner a bargain. Each prisoner is given the opportunity either to betray the other by testifying that the other committed the crime, or to cooperate with the other by remaining silent. The possible outcomes are:

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Behavioral game theory analyzes interactive strategic decisions and behavior using the methods of game theory, experimental economics, and experimental psychology. Experiments include testing deviations from typical simplifications of economic theory such as the independence axiom and neglect of altruism, fairness, and framing effects. As a research program, the subject is a development of the last three decades. Traditional game theory focuses on mathematical equilibriums, utility maximizing, and rational choice; in contrast, behavioral game theory focuses on choices made by participants in studies and is game theory applied to experiments. Choices studied in behavioral game theory are not always rational and do not always represent the utility maximizing choice.

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The mean contribution rate among the 60 Japanese subjects was 80%
The mean contribution rate among the 39 American subjects was 69%