Observations of individual subjects, game effects, and learning effects), by fitting

Observations of individual subjects, game effects, and learning effects), by fitting the data to the following nonnested, Sch66336 biological activity multilevel model (27): yi = treatment + s tsubject+ ggame+ hhistory+ i ;[1]ACKNOWLEDGMENTS. The MiransertibMedChemExpress Miransertib authors thank Andrew Gelman for advice on statistical modeling and Winter Mason and Daniel Goldstein for helpful conversations.Wang et al.PNAS | September 4, 2012 | vol. 109 | no. 36 |SOCIAL SCIENCESwhere yi is the expected cooperation level for the ith observation (i = 1, . . . nobs, and t[i], s[i], g[i], and h[i] are all index variables that map the ith observation to a particular treatment, subject, game, and experience level, respectively. Each observation refers to the average contribution of a particular player in a single game; hence for the initial payoffs, we have nobs = 82 ?24 = 1,968, and for the modified payoffs, nobs = 12 ?24 = 288). Moreover, treatment is a dummy variable for treatment, where t[i] = 1, . . . , 20 t for the initial payoffs (we have two initial conditions, a static case for each, and r = 1, 3, 6 and k = 1, 3, 5; hence 2 ?3 ?3 + 2 = 20 treatments), and for the modified payoffs t[i] = 1, . . . , 4 (we have one initial condition, a single static case, and r = 1 and k = 1, 5, 23, and hence 4 treatments); subject N ?; 2 subject ?s is a group-level predictor for subjects; game N ?; 2 ?is a group-level game g history predictor for games; and h N ?; 2 history ?is a group-level predictor for the number of games played by player s[i] at the time of the g[i]th game. Note that unlike for subject, game, and history effects, we do not model the treatment effects, preferring the simpler and more conservative approach of using a dummy variable for each treatment and hence avoiding the need to worry about potentially erroneous distributional assumptions (27). To test for significance, we computed 95 confidence intervals for the difference between each treatment and its corresponding static case; hence if zero is not contained in the interval, then the null hypothesis that they are the same can be rejected at the 5 level. As described in the main text, for the initial payoffs the null hypothesis can be rejected for all treatments except for r = 6 and k = 1, 3, 5 for the cliques initial conditions and r = 6 and k = 1 for the random initial condition. For the modified payoffs, all treatments had a significant and positive effect.1. Axelrod R, Hamilton WD (1981) The evolution of cooperation. Science 211:1390?1396. 2. Ledyard J (1995) Public goods: A survey of experimental research. The Handbook of Experimental Economics, eds Kagel JH, Roth AE (Princeton University Press, Princeton, NJ), pp 111?94. 3. Chaudhuri A (2011) Sustaining cooperation in laboratory public goods experiments: A selective survey of the literature. Exp Econ 14:47?3. 4. Fischbacher U, G hter S, Fehr E (2001) Are people conditionally cooperative? Evidence from a public goods experiment. Econ Lett 71:397?04. 5. Fehr E, G hter S (2000) Cooperation and punishment in public goods experiments. Am Econ Rev 90:980?94. 6. Sefton M, Shupp R, Walker J (2007) The effect of rewards and sanctions in provision of public goods. Econ Inq 45:671?90. 7. Kossinets G, Watts DJ (2006) Empirical analysis of an evolving social network. Science 311:88?0. 8. Fehl K, van der Post DJ, Semmann D (2011) Co-evolution of behaviour and social network structure promotes human cooperation. Ecol Lett 14:546?51. 9. Rand D, Arbesman S, Christakis N (2011) Dynamic social.Observations of individual subjects, game effects, and learning effects), by fitting the data to the following nonnested, multilevel model (27): yi = treatment + s tsubject+ ggame+ hhistory+ i ;[1]ACKNOWLEDGMENTS. The authors thank Andrew Gelman for advice on statistical modeling and Winter Mason and Daniel Goldstein for helpful conversations.Wang et al.PNAS | September 4, 2012 | vol. 109 | no. 36 |SOCIAL SCIENCESwhere yi is the expected cooperation level for the ith observation (i = 1, . . . nobs, and t[i], s[i], g[i], and h[i] are all index variables that map the ith observation to a particular treatment, subject, game, and experience level, respectively. Each observation refers to the average contribution of a particular player in a single game; hence for the initial payoffs, we have nobs = 82 ?24 = 1,968, and for the modified payoffs, nobs = 12 ?24 = 288). Moreover, treatment is a dummy variable for treatment, where t[i] = 1, . . . , 20 t for the initial payoffs (we have two initial conditions, a static case for each, and r = 1, 3, 6 and k = 1, 3, 5; hence 2 ?3 ?3 + 2 = 20 treatments), and for the modified payoffs t[i] = 1, . . . , 4 (we have one initial condition, a single static case, and r = 1 and k = 1, 5, 23, and hence 4 treatments); subject N ?; 2 subject ?s is a group-level predictor for subjects; game N ?; 2 ?is a group-level game g history predictor for games; and h N ?; 2 history ?is a group-level predictor for the number of games played by player s[i] at the time of the g[i]th game. Note that unlike for subject, game, and history effects, we do not model the treatment effects, preferring the simpler and more conservative approach of using a dummy variable for each treatment and hence avoiding the need to worry about potentially erroneous distributional assumptions (27). To test for significance, we computed 95 confidence intervals for the difference between each treatment and its corresponding static case; hence if zero is not contained in the interval, then the null hypothesis that they are the same can be rejected at the 5 level. As described in the main text, for the initial payoffs the null hypothesis can be rejected for all treatments except for r = 6 and k = 1, 3, 5 for the cliques initial conditions and r = 6 and k = 1 for the random initial condition. For the modified payoffs, all treatments had a significant and positive effect.1. Axelrod R, Hamilton WD (1981) The evolution of cooperation. Science 211:1390?1396. 2. Ledyard J (1995) Public goods: A survey of experimental research. The Handbook of Experimental Economics, eds Kagel JH, Roth AE (Princeton University Press, Princeton, NJ), pp 111?94. 3. Chaudhuri A (2011) Sustaining cooperation in laboratory public goods experiments: A selective survey of the literature. Exp Econ 14:47?3. 4. Fischbacher U, G hter S, Fehr E (2001) Are people conditionally cooperative? Evidence from a public goods experiment. Econ Lett 71:397?04. 5. Fehr E, G hter S (2000) Cooperation and punishment in public goods experiments. Am Econ Rev 90:980?94. 6. Sefton M, Shupp R, Walker J (2007) The effect of rewards and sanctions in provision of public goods. Econ Inq 45:671?90. 7. Kossinets G, Watts DJ (2006) Empirical analysis of an evolving social network. Science 311:88?0. 8. Fehl K, van der Post DJ, Semmann D (2011) Co-evolution of behaviour and social network structure promotes human cooperation. Ecol Lett 14:546?51. 9. Rand D, Arbesman S, Christakis N (2011) Dynamic social.