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

A cooperative game with envy

Jiahua Zhang 1, , Shu-Cherng Fang 2, , Yifan Xu 1, and  Ziteng Wang 3,,

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.

Keywords: Cooperative game, core, envy, coalition, behavior.
Mathematics Subject Classification: Primary: 91A12, 91A30.
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. Google Scholar

[2]

R. Aumann and J. Dreze, Cooperative games with coalition structures, International Journal of Game Theory, 3 (1974), 217-237. doi: 10.1007/BF01766876. Google Scholar

[3]

J. N. Bearden, Ultimatum Bargaining Experiments: The State of the Art, Working paper, University of North Carolina, Chapel Hill, 2001.Google Scholar

[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. Google Scholar

[5]

G. E. Bolton, A comparative model of bargaining: Theory and evidence, The American Economic Review, 81 (1991), 1096-1136. Google Scholar

[6]

O. N. Bondareva, Some applications of linear programming methods to the theory of cooperative games (Russian), Problemy Kybernetiki, 10 (1963), 119-139. Google Scholar

[7]

C. Camerer, Behavioral Game Theory: Experiments in Strategic Interaction, Princeton, NJ: Princeton University Press, 2003.Google Scholar

[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. Google Scholar

[9]

M. Carter, Foundations of Mathematical Economics, Cambridge, London: MIT Press,Google Scholar

[10]

Y. Chen and S. X. Li, Group identity and social preferences, The American Economic Review, 99 (2009), 431-457. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[14]

E. Fehr and K. M. Schmidt, A theory of fairness, competition, and cooperation, The Quarterly Journal of Economics, 114 (1999), 817-868. Google Scholar

[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. Google Scholar

[16]

R. H. Frank, Choosing the Right Pond: Human Behavior and the Quest for Status, New York: Oxford University Press, 2003.Google Scholar

[17]

Some Theorems on N-person Games, PhD thesis, Princeton, NJ: Princeton University, 1953. Google Scholar

[18]

P. J. Hammond, Envy, In: J. Eatwall, M. Milgate, P. Newman (eds) The New Palgrave Dictionary of Economics, New York: Stockton Press, 1987.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[25]

M. Montero, Inequity aversion may increase inequity, The Economic Journal, 117 (2007), C192-C204. Google Scholar

[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. Google Scholar

[27]

M. J. Osborne and A. Rubinstein, A Course in Game Theory, London: MIT Press, 1994. Google Scholar

[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. Google Scholar

[29]

B. Peleg and P. Sudhölter, Introduction to the Theory of Cooperative Games, 2nd edn., Springer US, 2007. Google Scholar

[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. Google Scholar

[31]

A. Rees, The role of fairness in wage determination, Journal of Labor Economics, 11 (1993), 243-252. doi: 10.1086/298325. Google Scholar

[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. Google Scholar

[33]

H. E. Scarf, The core of an N person game, Econometrica, 35 (1967), 50-69. doi: 10.2307/1909383. Google Scholar

[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. Google Scholar

[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. Google Scholar

[36]

L. S. Shapley, On balanced sets and cores, Naval Research Logistics Quarterly, 14 (1967), 453-460. doi: 10.1002/nav.3800140404. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[40]

R. Vecchio, Explorations in employee envy: Feeling envious and feeling envied, Cognition & Emotion, 19 (2010), 69-81. doi: 10.1080/02699930441000148. Google Scholar

[41]

S. S. Yi, Stable coalition structures with externalities, Games and Economic Behavior, 20 (1997), 201-237. doi: 10.1006/game.1997.0567. Google Scholar

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. Google Scholar

[2]

R. Aumann and J. Dreze, Cooperative games with coalition structures, International Journal of Game Theory, 3 (1974), 217-237. doi: 10.1007/BF01766876. Google Scholar

[3]

J. N. Bearden, Ultimatum Bargaining Experiments: The State of the Art, Working paper, University of North Carolina, Chapel Hill, 2001.Google Scholar

[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. Google Scholar

[5]

G. E. Bolton, A comparative model of bargaining: Theory and evidence, The American Economic Review, 81 (1991), 1096-1136. Google Scholar

[6]

O. N. Bondareva, Some applications of linear programming methods to the theory of cooperative games (Russian), Problemy Kybernetiki, 10 (1963), 119-139. Google Scholar

[7]

C. Camerer, Behavioral Game Theory: Experiments in Strategic Interaction, Princeton, NJ: Princeton University Press, 2003.Google Scholar

[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. Google Scholar

[9]

M. Carter, Foundations of Mathematical Economics, Cambridge, London: MIT Press,Google Scholar

[10]

Y. Chen and S. X. Li, Group identity and social preferences, The American Economic Review, 99 (2009), 431-457. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[14]

E. Fehr and K. M. Schmidt, A theory of fairness, competition, and cooperation, The Quarterly Journal of Economics, 114 (1999), 817-868. Google Scholar

[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. Google Scholar

[16]

R. H. Frank, Choosing the Right Pond: Human Behavior and the Quest for Status, New York: Oxford University Press, 2003.Google Scholar

[17]

Some Theorems on N-person Games, PhD thesis, Princeton, NJ: Princeton University, 1953. Google Scholar

[18]

P. J. Hammond, Envy, In: J. Eatwall, M. Milgate, P. Newman (eds) The New Palgrave Dictionary of Economics, New York: Stockton Press, 1987.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[25]

M. Montero, Inequity aversion may increase inequity, The Economic Journal, 117 (2007), C192-C204. Google Scholar

[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. Google Scholar

[27]

M. J. Osborne and A. Rubinstein, A Course in Game Theory, London: MIT Press, 1994. Google Scholar

[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. Google Scholar

[29]

B. Peleg and P. Sudhölter, Introduction to the Theory of Cooperative Games, 2nd edn., Springer US, 2007. Google Scholar

[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. Google Scholar

[31]

A. Rees, The role of fairness in wage determination, Journal of Labor Economics, 11 (1993), 243-252. doi: 10.1086/298325. Google Scholar

[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. Google Scholar

[33]

H. E. Scarf, The core of an N person game, Econometrica, 35 (1967), 50-69. doi: 10.2307/1909383. Google Scholar

[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. Google Scholar

[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. Google Scholar

[36]

L. S. Shapley, On balanced sets and cores, Naval Research Logistics Quarterly, 14 (1967), 453-460. doi: 10.1002/nav.3800140404. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[40]

R. Vecchio, Explorations in employee envy: Feeling envious and feeling envied, Cognition & Emotion, 19 (2010), 69-81. doi: 10.1080/02699930441000148. Google Scholar

[41]

S. S. Yi, Stable coalition structures with externalities, Games and Economic Behavior, 20 (1997), 201-237. doi: 10.1006/game.1997.0567. Google Scholar

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