In Partial Fulfilbnent of the Requirements for thE' Degree of

Each of the three independent chapters of this dissertation examines or justifies cooperative behavior in one of two specific public goods settings. Game duration is a treatment variable; Experiments simulating games with finite and infinite repetitions and conducted.

List of Figures

Chapter 1

An Experimental Analysis of Two-Person Reciprocity

Games

Recipr o c i ty Games
Related Research
The Experimental Design

Equilibria

The Experiments
The Results

The One-Shot Treatment
The Finite and Infinite Repetition Treatments

However, in G3 and G4 the finite iteration creates many additional equilibria. The popular finite game theorem, any maximally dominating outcome can be approximated by a perfect equilibrium under.with if the number of. the phase is large enough.3 The result of the popular theorem causes a problem that. is very similar to the problem encountered in infinitely repeated games of how players coordinate in a given equilibrium when the set of equilibria is very large. After the 15th period, a ten-sided cartridge was rolled up so that subjects could catch the outcome.

Average Payoffs

Comparing Average Payoffs

In the one-shot treatment, the average ncar payoff is associated with the Nash Solution, which assigns each player four. The average column player's payoff drops more than 20 percent when moving from the one-shot treatment to either the finite or infinite repetition treatment of (/4.

The Finite and Infinite Repetition Treatments

From the group's perspective, this drop in the column player's payoff is not offset by the small increase in the row player's payoff. The average lineman only gets about 10 [Wrccnt more when moving from the one shot, or repeated trcatment of G.1.

The Strategy Space

Conclusions

If you are interested in the total payouts, you can choose to shorten the length of the game. On the other hand, the results of symmetric games are very encouraging from a policy point of view.

Sample Instructions and Quiz

INSTRUCTIONS FOR A DECISION-MAKING EXPERIMENT

If you choose <"' option A and your partner chooses option C, you get l points. 3 ROWS 7 to 9: If you choose option C, you get 1 point if your partner chooses either option A or option B.

Quiz

At the end of the experiment, you will be paid 5 cents for each point collected.

QUIZ

Tables

We compare the distribution of strategies when there is no punishment strategy with the distribution of strategies when a. We compare the distribution of strategies when there is no punishment strategy with the distribution of strategies when there is.

Figures

Chapter 2

A Bounded Rationality, Evolutionary Model for

Population Games and the Replicator D y namic

Each player in the population lives (plays a game), creates offspring identical to itself, and then dies. In Replicator Dynamic, each player produces a number of offspring proportional to that player's lifetime fitness or payout. So, if the Replicator dynamic is ever in equilibrium, meaning that the mix of player types is the same from generation to generation, the strategies that remain in equilibrium have been justified in a Darwinian sense.

At these tournaments, several people, mostly professional scientists, submitted computer programs that were essentially strategies in the iterated Prisoner's Dilemma. Boyd and Lorberbaum showed that, contrary to previous optimistic research, no pure strategy is evolutionarily stable in the infinitely repeated Prisoner's Dilemma.

The Environment

Another way to think about the set is the set of strategies that can be implemented by a two-state automaton, such automata are usually called Moore machines. This convention allows a machine to be written as a triple, for example {a, a, a}, where the first qo represents the machine's initial move, the second represents >.(a), the move the machine chooses if its opponent chooses action a., and the third represents >.(b), the action chosen if his opponent chooses action b. Machine {a, a, a} plays action 'a' on the first move and then plays action 'a' regardless of what action its opponent chooses.

Ultimately, due to the finite strategies, the order of the game is cyclical, with the longest cycle comprising four phases. The application of the constraint of bounded rationality and the specific definition of the payout functions have transformed the infinitely iterated game.

The Replicator

Other Symmetric Garnes
Asymmetric Garnes

The assumption means that <'Most possible strategy has at least some representation in the population. A particularly mixed strategy will be clenot<'d q; and will be treated in the obvious way by the function F. It is also a subgame-perfect Nash equilibrium in the game G00, even though every other equilibrium has Pareto Superior payoffs.

Similarly, Lemma 4 can be used to show that Ps and then P3 and P4 go to zero as t goes to infinity; every time qe the mixed strategy is ceded in the Lemma. No matter what the initial population is (as long as it is in the interior of ~), the Rcplicator Dynamic will converge to an equilibrium with only s5 and s6 players.

Examples

This method is an improvement over the two-population method because it does not change the outcome predicted in the Reciprocity Game. Unfortunately, even with random population allocation, the Dattle of the Sexes does not satisfy Assumption 2. In equilibrium, there are only three possible outcomes for an encounter between two players, they call: Alternation, Dominant Strategy Nash Play, and Irrational Games.

Dominant Strategy Nash Play occurs when a player with strategy 8 6 meets another player with strategy s6. As an example of what happens if the payoff matrix is not constructed with the correct inequalities, consider the payoff matrix M2.

2. 6 References

Tables

Figures

Phase Portrait for Example

Phase Portrait for Counter-Example

Initial Generation B

Chapter 3

Anomalous Behavior in Linear Public Goods Experiments

How Much and Why?

Introduction

We present the following thought experiment in the context of a well-studied allocation mechanism for private goods, the second-price auction, in the hope that it will help the reader understand some of our concerns about design, and to foreshadow what follows . The first observation to be made about the thought experiment is that it shares some of the characteristics of many voluntary contribution public goods experiments reported in the literature. In the most common voluntary contribution public goods experiment, as in the thought experiment, there are a number of identical players.

In an auction, each player assigns the same value to the good in each of the ten auctions. None of these experiments varied subjects' individual preferences between decisions, nor did they provide explicit information about the distribution of preferences in the population.

Background

One advantage is simply better measurement: response functions (bid functions) can be estimated on an individual level. Several findings emerged from these other studies: (1) almost all players in this game violate their one-shot dominant strategy, with many contributing more than half of their endowment, even if r;/V is three or more; (2) there is a strong negative relationship between the marginal replacement rate r;/V and the rate at which violations are observed; (3) about half of the total private capital is contributed by inexperienced subjects at the first game of the game; (4) violations of dominant strategies decrease with repetition and with experience (playing a second set of games with a new group); (.5) Violations of dominant strategies to contribute (rJV; < 1, Saijo and Yamaguchi [1992]) appear to be even more common than violations of dominant strategies to piggyback.

Our Design and Procedures

Therefore, the data contain multiple observations of each individual's choice behavior, observations at different levels of r;jV, and allow estimation of response functions at both the individual and aggregate levels. One of the distributions, V = 3, has the property that group efficiency is not maximized when all subjects contribute in each round. In this situation, on average, forty percent of the time subjects are assigned a property value that is worth more than four times the individual marginal value of the public good.

This allows direct comparison with some past experiments, particularly those reported in the Isaac and Walker studies. Each subject was paid in cash at a session-specific exchange rate for each point earned in the session.

3 .4 Response Functions and Backgroun d Noise

Analysis of the data

Some baselines

Due to the linear structure of the environment, such behavior is not rational, even if a subject is given a warm glow at its marginal rate of substitution. In other words, virtually all splits can be explained by subjects using a dominant strategy of free ride. Many have speculated that subjects violate their dominant piggyback strategy because of some form of altruism, or vice versa, because their utility function positively depends on group payoffs.

In our experiments, four percent of decisions violate the dominant strategy of contributing when 1·;/V < 1. As Table 7 shows, this kind of behavior is about as common as malicious behavior, but practically disappears with experience (1 observation out of 129).

Estimation of response functions from aggre- gate data

Response Functions and Errors: Individual Level Analysis

A second thing to note is that not every subject has the same estimated cut off. In Table 9, for example, subject #2 has an estimated intercept of 2.17 (corresponding to a sign value of 13) while subject #16 has an estimated intercept of 1.0 (corresponding to a sign value of 6). On tlw :r-axis is the difference between the estimated intercept and the value of the public good in tok(a value units.

For example, topic #1 from Table 10 would be included in the category in this figure because its estimated cutpoint is 9 and the value of the public good is 6. Finally, we define consistent players as players who can be classified perfectly. , so they never make a mistake on their estimated border point.

3 .5.4 Comparison to Previous R esults

Interpreting the Results

The findings, which repeat from previous experiments with comparable group sizes, are that experience leads to lower contribution rates, and contribution rates fall in the marginal rate of substitution (marginal valuation of the private good). On the environmental side, our experiments use a different economic environment, by which we mean that the information structure and the profile of preferences in the group differ. In particular, as emphasized in the introduction, the information structure and profile of preferences correspond almost exactly to the standard environment used for auction experiments.

In each period, group preferences are randomly and independently drawn from a known distribution of marginal rates of substitution, thereby introducing heterogeneity among individuals. To determine the effects of tlw private information in our experiments, we conducted two open-information sessions (with V = 6 and X = 9) where all token value draws were revealed to everyone in the group.

Sample Instructions

Decis ion-Making Exp e riment

RULES FOR EXPERIMENT #1

PAYOFFS