Cooperation in an Assortative Matching Prisoners Dilemma Experiment with Pro-Social Dummies

Assortative matching (AM) can be theoretically an effective means to facilitate cooperation. We designed a controlled lab experiment with three treatments on multi-round prisoner’s dilemma. With matching based on weighted history (WH) as surrogate for AM, we show that adding pro-social dummies to the WH treatment may significantly improve cooperation, compared to both the random matching and the WH treatment. In society where assortative matching is effective and promoted by the underlying culture, institutional promotion of virtue role models can be interpreted as generating additional pro-social dummies, so as to move the initial state of cooperators into the basin of attraction for a highly cooperative polymorphic equilibrium.

given during the experiment.) In each round, when the decision bars become highlighted, you can start to make a decision between A and B. An example is shown in these printed instructions to tell you how to find out what you and your counterpart will receive as a consequence of your decisions.
At the end, you will be paid off the sum of the payoffs you get in each round, plus a 50NT show-up fee. For experimental purposes, in some of the games you will not be informed immediately of the action of the other person and thus of your own payoff there. On the upper right of the window, you will see your account balance for all the other games/rounds played, including the 50 NT show-up fee. At the end, your final balance will be updated, with all the payoff and action information previously withheld.
Note that the computer rematches the participants into pairs after each round. The matching procedure changes during the session. You will be informed about the details in special written instructions accordingly. Wait for the experimenter's further instruction.
During the session, you will find the following information on the right of the window: (1) Your own choices so far; (2) Your counterpart's choices; (3) Your own payoffs in the previous rounds; (4) Other information and instructions. For part of the experiment, you will not get information about (2) and (3) right away, but you will find "?" instead. This information will be given at the end of the session.

Special Instructions
Random Matching: 5 rounds no feedback (Games 1 and 3) In this part, there will be five rounds. At the beginning of each round, subjects will be randomly paired to play the game you will find on the window. This means that you have the same chance to meet any of the other 13 subjects in each round, independent of what has happened so far. For experimental purposes, you will be informed neither of your counterpart's action nor of your own payoff, at this time. You will receive the relevant information at the end of the session and your total final payoff will be updated accordingly.
Random Matching: 25 rounds (Game 2) In this part, there will be 25 rounds. At the beginning of each round, subjects will be randomly paired to play the game you will find on the window. This means that you have the same chance to meet any of the other 13 subjects in each round, independent of what has happened so far. For illustration, assume that somebody's choices in round 1-6 are ABBAABA.

Weighted
For example, in round 7, T=5 is calculated in the following way. Since A is the choice in the previous round (round 6), T1=0. B in the 2 nd previous one (round 5), T2=3. Etc.
Step 2: Matching according to the sorting scores At the beginning of each round, all subjects will be ranked according to their ranking scores T and be paired with their neighbors, subsequently from top to bottom.
Subjects with the same ranking score T will be ranked randomly among themselves. Note that this matching procedure ignores the actions earlier than five rounds ago.
Also you can at any time find out in the window on the right side of your monitor about your own previous choices, your previous counterpart's choices, your payoffs, your account balance, and your current ranking score T.
3. If six subjects a, b, c, d, e, f participate in the experiment and, at some round, have the ranking scores of 8, 12, 10, 12, 5, 0, then a can be potentially matched with: (1) only b, (2)         Note: Except for RM, it seems that Game 2 (rounds 1~5) has a stronger correlation than Game 1, with Game 2 (rounds 6~23). Both Game 1 and Game 2 (rounds 1~5) can be used as explanatory variable in P(C) in Game 2 (rounds 6~23). It seems that Game 2 (rounds 1~5) would have higher impact in WH treatments. Figure A4. Probability of cooperation in Game 2 Rounds 1~5 vs. Rounds 6~23 We observe in Table 6.2 that in both of #C=4 and =5 groups some treatments have too small sizes of sample. Merging the two classes yields numbers between 8 and 11 that matches the smallest in all other classes.
2  -tests that this distribution is not significantly different also on the individual level with N=70 for each sample, p = .2704, as additional evidence for non-bias in sampling. Figure A5. The probability of cooperation vs. matching score T in Game 2

General Notes on Logistic Regression
Logistic regression is a regression method used to deal with the case where the response (say, variable y) is dichotomous. Since the response is either 0 or 1, the logistic regression is to model the probability of y = 1 as ) exp( ) , , | 1 ( where k x x , , 1  are independent variables. The logistic regression is usually conducted via logarithm of odds, or where the odds is defined as ) , The significance of independent variable i x can be judged by odds ratio, or ) exp( i  . However, like in the linear regression, the model with the largest R 2 is not necessarily the best model and there are no unique measures to determine the best model in logistic regression. Still, there are some measures which can be used to choose feasible models in logistic regression. We choose two frequently measures: logarithm of likelihood and concordance, and larger values of these two measures indicate a better fit. We can also use the logarithm of likelihood and the number of parameters, to avoid the possibility of over-fit (i.e., using too many independent variables). In addition, we shall check if the model assumption of logistic regression is violated. A model with larger logarithm of likelihood and concordance, provided that the model assumption is not violated, is preferred. Note that the model assumption is usually evaluated by the goodness-of-fit test.
Two frequently used goodness-of-fit are Pearson and Deviance tests. However, as the number of groups increases, these two tests are more likely to falsely reject the null hypothesis. Another (Pearson-like) goodness-of-fit test, proposed by Hosmer and Lemeshow (1980), is used more often in practice, as in this paper. Note that the Hosmer-Lemeshow (H-L) test is to group residuals based on the values of the estimated probabilities. The default number of groups in H-L test is 10, as in this paper.

Regressions for the paper
Forms for the models considered are as follows, where I denotes indicator function. We further compare the parameters for different treatments, as shown in Table A7.1.
The values of parameters 0 , 1 , 2 , 3 are similar for RM and WH, and the major difference is on 1 . It seems that the players for RM with higher cooperation level in Game 1 would have higher cooperation level in Game 2, comparing to those for WH.
The players for WHc behave quite differently. (WHc has clearly higher 0 , which means lower initial inclination to play D, compared to RM and WH.) Note: Shaded rows indicate best fitting models in respective treatments. Boldface indicates failed goodness-of-fit tests. The model "Game 1" is the best for RM and WH. "Game 2 (t=1-5)" would be best for WHc, if not for its HL-test failure. Yet, replacing it with the dummy, "Game 2 (#C=1,4,5)" gets it done.