# American Institute of Mathematical Sciences

October  2017, 13(4): 2049-2066. doi: 10.3934/jimo.2017031

## A cooperative game with envy

 1 School of Management, Fudan University, Shanghai 200433, China 2 Edward P. Fitts Department of Industrial and Systems Engineering, North Carolina State University, Raleigh, NC 27695, USA 3 Department of Industrial and Systems Engineering, Northern Illinois University, DeKalb, IL 60115, USA

* Corresponding author: Ziteng Wang

Received  January 2016 Revised  September 2016 Published  April 2017

This paper proposes an envy-incorporated cooperative game model and investigates the envy effects on players' coalition-forming and payoff-allocating decisions. A player's utility depends on her/his own payoff and the envy toward other players, inside and outside the same coalition, with higher payoffs. Envy core is defined to characterize stable coalition structures and payoff allocations of this new game. Conditions for the envy core to be nonempty are provided. The relative significance of in-coalition envy and out-coalition envy is shown to be a key factor to the form of the envy core. Application to a simple game shows that envy may significantly change players' decisions.

Citation: Jiahua Zhang, Shu-Cherng Fang, Yifan Xu, Ziteng Wang. A cooperative game with envy. Journal of Industrial & Management Optimization, 2017, 13 (4) : 2049-2066. doi: 10.3934/jimo.2017031
##### References:
 [1] G. A. Akerlof and R. E. Kranton, Economics and identity, The Quarterly Journal of Economics, 115 (2000), 715-753. doi: 10.1515/9781400834181. [2] R. Aumann and J. Dreze, Cooperative games with coalition structures, International Journal of Game Theory, 3 (1974), 217-237. doi: 10.1007/BF01766876. [3] J. N. Bearden, Ultimatum Bargaining Experiments: The State of the Art, Working paper, University of North Carolina, Chapel Hill, 2001. [4] L. J. Billera, Some theorems on the core of an n-person game without side-payments, SIAM Journal on Applied Mathematics, 18 (1970), 567-579. doi: 10.1137/0118049. [5] G. E. Bolton, A comparative model of bargaining: Theory and evidence, The American Economic Review, 81 (1991), 1096-1136. [6] O. N. Bondareva, Some applications of linear programming methods to the theory of cooperative games (Russian), Problemy Kybernetiki, 10 (1963), 119-139. [7] C. Camerer, Behavioral Game Theory: Experiments in Strategic Interaction, Princeton, NJ: Princeton University Press, 2003. [8] L. A. Cameron, Raising the stakes in the ultimatum game: Experimental evidence from Indonesia, Economic Inquiry, 37 (1999), 47-59. doi: 10.1111/j.1465-7295.1999.tb01415.x. [9] M. Carter, Foundations of Mathematical Economics, Cambridge, London: MIT Press, [10] Y. Chen and S. X. Li, Group identity and social preferences, The American Economic Review, 99 (2009), 431-457. [11] M. K. Duffy, J. D. Shaw and J. M. Schaubroeck, Envy in organizational life, In: R. Smith (ed) Envy: Theory and Research, New York: Oxford University Press, 2008. doi: 10.1093/acprof:oso/9780195327953.003.0010. [12] B. Dutta and D. Ray, Constrained egalitarian allocations, Games and Economic Behavior, 3 (1991), 403-422. doi: 10.1016/0899-8256(91)90012-4. [13] C. L. Eavey and G. J. Miller, Fairness in majority rule games with a core, American Journal of Political Science, 28 (1984), 570-586. doi: 10.2307/2110905. [14] E. Fehr and K. M. Schmidt, A theory of fairness, competition, and cooperation, The Quarterly Journal of Economics, 114 (1999), 817-868. [15] D. S. Felsenthal, D. Leech, C. List and M. Machover, In defence of voting power analysis: Responses to Albert, European Union Politics, 4 (2003), 473-497. doi: 10.1177/146511650344005. [16] R. H. Frank, Choosing the Right Pond: Human Behavior and the Quest for Status, New York: Oxford University Press, 2003. [17] Some Theorems on N-person Games, PhD thesis, Princeton, NJ: Princeton University, 1953. [18] P. J. Hammond, Envy, In: J. Eatwall, M. Milgate, P. Newman (eds) The New Palgrave Dictionary of Economics, New York: Stockton Press, 1987. [19] J. Henrich, Does culture matter in economic behavior? Ultimatum game bargaining among the Machiguenga of the Peruvian Amazon, The American Economic Review, 90 (2000), 973-979. [20] J. P. Kahan and A. Rapoport, Coalition formation in the triad when two are weak and one is strong, Mathematical Social Sciences, 1 (1980), 11-37. doi: 10.1016/0165-4896(80)90003-7. [21] H. Keiding and L. Thorlund-Petersen, The core of a cooperative game without side payments, Journal of Optimization Theory & Applications, 54 (1987), 273-288. doi: 10.1007/BF00939435. [22] G. Kirchsteiger, The role of envy in ultimatum games, Journal of Economic Behavior & Organization, 25 (1994), 373-389. doi: 10.1016/0167-2681(94)90106-6. [23] S. Kohler, Envy can promote more equal division in alternating-offer bargaining, Journal of Neuroscience, Psychology, and Economics, 6 (2013), 31-41. doi: 10.1037/npe0000001. [24] J. Lee and T. S. Driessen, Sequentially two-leveled egalitarianism for TU games: Characterization and application, European Journal of Operational Research, 220 (2012), 736-743. doi: 10.1016/j.ejor.2012.02.014. [25] M. Montero, Inequity aversion may increase inequity, The Economic Journal, 117 (2007), C192-C204. [26] A. Okada and A. Riedl, Inefficiency and social exclusion in a coalition formation game: Experimental evidence, Games and Economic Behavior, 50 (2005), 278-311. doi: 10.1016/j.geb.2004.02.006. [27] M. J. Osborne and A. Rubinstein, A Course in Game Theory, London: MIT Press, 1994. [28] W. G. Parrott and R. H. Smith, Distinguishing the experiences of envy and jealousy, Journal of Personality and Social Psychology, 64 (1993), 906-920. doi: 10.1037/0022-3514.64.6.906. [29] B. Peleg and P. Sudhölter, Introduction to the Theory of Cooperative Games, 2nd edn., Springer US, 2007. [30] A. Predtetchinski and P. J. J. Herings, A necessary and sufficient condition for non-emptiness of the core of a non-transferable utility game, Journal of Economic Theory, 116 (2004), 84-92. doi: 10.1016/S0022-0531(03)00261-8. [31] A. Rees, The role of fairness in wage determination, Journal of Labor Economics, 11 (1993), 243-252. doi: 10.1086/298325. [32] A. E. Roth, V. Prasnikar, M. Okuno-Fujiwara and S. Zamir, Bargaining and market behavior in Jerusalem, Ljubljana, Pittsburgh, and Tokyo: An experimental study, The American Economic Review, 81 (1991), 1068-1095. [33] H. E. Scarf, The core of an N person game, Econometrica, 35 (1967), 50-69. doi: 10.2307/1909383. [34] J. J. Seta and C. E. Seta, Big fish in small ponds: A social hierarchy analysis of intergroup bias, Journal of personality and social psychology, 71 (1996), 1210-1221. doi: 10.1037/0022-3514.71.6.1210. [35] L. S. Shapley, A value for n-person games, In: Kuhn HW, Tucker AW (eds) Contributions to the Theory of Games, Princeton, NJ: Princeton University Press, 2 (1953), 307-317. [36] L. S. Shapley, On balanced sets and cores, Naval Research Logistics Quarterly, 14 (1967), 453-460. doi: 10.1002/nav.3800140404. [37] L. S. Shapley, On balanced games without side payments, In: Hu TC, Robinson SM (eds) Mathematical Programming, New York: Academic Press, 1973,261-290. [38] L. S. Shapley and M. Shubik, Quasi-cores in a monetary economy with nonconvex preferences, Econometrica: Journal of the Econometric Society, 34 (1966), 805-827. doi: 10.2307/1910101. [39] H. Tajfel and J. C. Turner, The social identity theory of intergroup behaviour, In: W. G. Austin, S. Worchel (eds), Psychology of Intergroup Relations (2nd Edition), Chicago: Nelson-Hall, 1986. [40] R. Vecchio, Explorations in employee envy: Feeling envious and feeling envied, Cognition & Emotion, 19 (2010), 69-81. doi: 10.1080/02699930441000148. [41] S. S. Yi, Stable coalition structures with externalities, Games and Economic Behavior, 20 (1997), 201-237. doi: 10.1006/game.1997.0567.

show all references

##### References:
 [1] G. A. Akerlof and R. E. Kranton, Economics and identity, The Quarterly Journal of Economics, 115 (2000), 715-753. doi: 10.1515/9781400834181. [2] R. Aumann and J. Dreze, Cooperative games with coalition structures, International Journal of Game Theory, 3 (1974), 217-237. doi: 10.1007/BF01766876. [3] J. N. Bearden, Ultimatum Bargaining Experiments: The State of the Art, Working paper, University of North Carolina, Chapel Hill, 2001. [4] L. J. Billera, Some theorems on the core of an n-person game without side-payments, SIAM Journal on Applied Mathematics, 18 (1970), 567-579. doi: 10.1137/0118049. [5] G. E. Bolton, A comparative model of bargaining: Theory and evidence, The American Economic Review, 81 (1991), 1096-1136. [6] O. N. Bondareva, Some applications of linear programming methods to the theory of cooperative games (Russian), Problemy Kybernetiki, 10 (1963), 119-139. [7] C. Camerer, Behavioral Game Theory: Experiments in Strategic Interaction, Princeton, NJ: Princeton University Press, 2003. [8] L. A. Cameron, Raising the stakes in the ultimatum game: Experimental evidence from Indonesia, Economic Inquiry, 37 (1999), 47-59. doi: 10.1111/j.1465-7295.1999.tb01415.x. [9] M. Carter, Foundations of Mathematical Economics, Cambridge, London: MIT Press, [10] Y. Chen and S. X. Li, Group identity and social preferences, The American Economic Review, 99 (2009), 431-457. [11] M. K. Duffy, J. D. Shaw and J. M. Schaubroeck, Envy in organizational life, In: R. Smith (ed) Envy: Theory and Research, New York: Oxford University Press, 2008. doi: 10.1093/acprof:oso/9780195327953.003.0010. [12] B. Dutta and D. Ray, Constrained egalitarian allocations, Games and Economic Behavior, 3 (1991), 403-422. doi: 10.1016/0899-8256(91)90012-4. [13] C. L. Eavey and G. J. Miller, Fairness in majority rule games with a core, American Journal of Political Science, 28 (1984), 570-586. doi: 10.2307/2110905. [14] E. Fehr and K. M. Schmidt, A theory of fairness, competition, and cooperation, The Quarterly Journal of Economics, 114 (1999), 817-868. [15] D. S. Felsenthal, D. Leech, C. List and M. Machover, In defence of voting power analysis: Responses to Albert, European Union Politics, 4 (2003), 473-497. doi: 10.1177/146511650344005. [16] R. H. Frank, Choosing the Right Pond: Human Behavior and the Quest for Status, New York: Oxford University Press, 2003. [17] Some Theorems on N-person Games, PhD thesis, Princeton, NJ: Princeton University, 1953. [18] P. J. Hammond, Envy, In: J. Eatwall, M. Milgate, P. Newman (eds) The New Palgrave Dictionary of Economics, New York: Stockton Press, 1987. [19] J. Henrich, Does culture matter in economic behavior? Ultimatum game bargaining among the Machiguenga of the Peruvian Amazon, The American Economic Review, 90 (2000), 973-979. [20] J. P. Kahan and A. Rapoport, Coalition formation in the triad when two are weak and one is strong, Mathematical Social Sciences, 1 (1980), 11-37. doi: 10.1016/0165-4896(80)90003-7. [21] H. Keiding and L. Thorlund-Petersen, The core of a cooperative game without side payments, Journal of Optimization Theory & Applications, 54 (1987), 273-288. doi: 10.1007/BF00939435. [22] G. Kirchsteiger, The role of envy in ultimatum games, Journal of Economic Behavior & Organization, 25 (1994), 373-389. doi: 10.1016/0167-2681(94)90106-6. [23] S. Kohler, Envy can promote more equal division in alternating-offer bargaining, Journal of Neuroscience, Psychology, and Economics, 6 (2013), 31-41. doi: 10.1037/npe0000001. [24] J. Lee and T. S. Driessen, Sequentially two-leveled egalitarianism for TU games: Characterization and application, European Journal of Operational Research, 220 (2012), 736-743. doi: 10.1016/j.ejor.2012.02.014. [25] M. Montero, Inequity aversion may increase inequity, The Economic Journal, 117 (2007), C192-C204. [26] A. Okada and A. Riedl, Inefficiency and social exclusion in a coalition formation game: Experimental evidence, Games and Economic Behavior, 50 (2005), 278-311. doi: 10.1016/j.geb.2004.02.006. [27] M. J. Osborne and A. Rubinstein, A Course in Game Theory, London: MIT Press, 1994. [28] W. G. Parrott and R. H. Smith, Distinguishing the experiences of envy and jealousy, Journal of Personality and Social Psychology, 64 (1993), 906-920. doi: 10.1037/0022-3514.64.6.906. [29] B. Peleg and P. Sudhölter, Introduction to the Theory of Cooperative Games, 2nd edn., Springer US, 2007. [30] A. Predtetchinski and P. J. J. Herings, A necessary and sufficient condition for non-emptiness of the core of a non-transferable utility game, Journal of Economic Theory, 116 (2004), 84-92. doi: 10.1016/S0022-0531(03)00261-8. [31] A. Rees, The role of fairness in wage determination, Journal of Labor Economics, 11 (1993), 243-252. doi: 10.1086/298325. [32] A. E. Roth, V. Prasnikar, M. Okuno-Fujiwara and S. Zamir, Bargaining and market behavior in Jerusalem, Ljubljana, Pittsburgh, and Tokyo: An experimental study, The American Economic Review, 81 (1991), 1068-1095. [33] H. E. Scarf, The core of an N person game, Econometrica, 35 (1967), 50-69. doi: 10.2307/1909383. [34] J. J. Seta and C. E. Seta, Big fish in small ponds: A social hierarchy analysis of intergroup bias, Journal of personality and social psychology, 71 (1996), 1210-1221. doi: 10.1037/0022-3514.71.6.1210. [35] L. S. Shapley, A value for n-person games, In: Kuhn HW, Tucker AW (eds) Contributions to the Theory of Games, Princeton, NJ: Princeton University Press, 2 (1953), 307-317. [36] L. S. Shapley, On balanced sets and cores, Naval Research Logistics Quarterly, 14 (1967), 453-460. doi: 10.1002/nav.3800140404. [37] L. S. Shapley, On balanced games without side payments, In: Hu TC, Robinson SM (eds) Mathematical Programming, New York: Academic Press, 1973,261-290. [38] L. S. Shapley and M. Shubik, Quasi-cores in a monetary economy with nonconvex preferences, Econometrica: Journal of the Econometric Society, 34 (1966), 805-827. doi: 10.2307/1910101. [39] H. Tajfel and J. C. Turner, The social identity theory of intergroup behaviour, In: W. G. Austin, S. Worchel (eds), Psychology of Intergroup Relations (2nd Edition), Chicago: Nelson-Hall, 1986. [40] R. Vecchio, Explorations in employee envy: Feeling envious and feeling envied, Cognition & Emotion, 19 (2010), 69-81. doi: 10.1080/02699930441000148. [41] S. S. Yi, Stable coalition structures with externalities, Games and Economic Behavior, 20 (1997), 201-237. doi: 10.1006/game.1997.0567.
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