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April  2022, 9(2): 151-163. doi: 10.3934/jdg.2022001

Properties of contests: Constructing contest success functions from best-responses

1. 

Department of Economics, Universidad Carlos Ⅲ, Madrid, Spain

2. 

Max Planck Institute for Tax Law and Public Finance, Munich, Bavaria, Germany

*Corresponding author: Luis Corchón

Received  February 2021 Revised  November 2021 Published  April 2022 Early access  January 2022

Fund Project: We thank Subhasish Chowdhury, two annonymous referees and the audience of Seminario de Dinámica Económica for U. de la República, Montevideo, Uruguay for helpful comments. All errors are our own. Luis Corchón acknowledges financial support from grants MDM 2014-0431, ECO2017_87769_P and S2015/HUM-3444

We aim at characterizing the functions that could be explained (recoverable) as a best reply of payoff-maximizing players in contests for a fixed prize. We show that recoverability strongly differs between Decisive Contests, where the prize is allocated with certainty, and Possibly Indecisive Contests, where the prize might not be awarded. In the latter, any arbitrary set of best reply functions is recoverable, thus "anything goes." In the former, best reply functions have to satisfy strong conditions in some cases. We provide an outline of possible applications of our results to R & D and labor markets.

Citation: Luis Corchón, Marco Serena. Properties of contests: Constructing contest success functions from best-responses. Journal of Dynamics and Games, 2022, 9 (2) : 151-163. doi: 10.3934/jdg.2022001
References:
[1]

D. Acemoglu and M. K. Jensen, Aggregate comparative statics, Games Econom. Behav., 81 (2013), 27-49.  doi: 10.1016/j.geb.2013.03.009.

[2]

J. A. Amegashie, A contest success function with a tractable noise parameter, Public Choice, 126 (2006), 135-144.  doi: 10.1007/s11127-006-2461-z.

[3]

A. Arad and S. Penczynski, Multi-dimensional reasoning in competitive resource allocation games: Evidence from intra-team communication, Mimeo, (2020). Available from: https://www.tau.ac.il/aradayal/MD_Reasoning_in_Resource_Allocation_Games.pdf.

[4]

K. H. Baik, Difference-form contest success functions and effort level in contests, European J. Political Economy, 14 (1998), 685-701.  doi: 10.1016/S0176-2680(98)00032-9.

[5]

A. Barge-Gil, E. Huergo, A. López and L. Moreno, Empirical models of firms' R & D, in Handbook of Game Theory and Industrial Organisation, Volume II, Edward Elgar, Chelthenam, UK; Northampton, MA, USA, 2018,475–516. doi: 10.4337/9781788112789.00027.

[6]

C. Beviá and and L. C. Corchón, Relative difference contest success function, Theory and Decision, 78 (2015), 377-398.  doi: 10.1007/s11238-014-9425-4.

[7]

P. R. Blavatskyy, Contest success function with the possibility of a draw: Axiomatization, J. Math. Econom., 46 (2010), 267-276.  doi: 10.1016/j.jmateco.2009.11.012.

[8]

Y.-K. Che and I. Gale, Difference-form contests and the robustness of all-pay auctions, Games Econom. Behav., 30 (2000), 22-43.  doi: 10.1006/game.1998.0709.

[9]

D. J. Clark and C. Riis, Contest success functions: An extension, Econom. Theory, 11 (1998), 201-204.  doi: 10.1007/s001990050184.

[10]

C. Cohen and A. Sela, Contests with ties, B. E. J. Theor. Econ., 7 (2007), 18pp. doi: 10.2202/1935-1704.1398.

[11]

L. Corchón, Theories of Imperfectly Competitive Markets, Springer, Berlin, Heidelberg, 2001. doi: 10.1007/978-3-662-04498-8.

[12]

L. Corchón and M. Dahm, Foundations for contest success functions, Econom. Theory, 43 (2010), 81-98.  doi: 10.1007/s00199-008-0425-x.

[13]

L. Corchón and M. Dahm, Welfare maximizing contest success functions when the planner cannot commit, J. Math. Econom., 47 (2011), 309-317.  doi: 10.1016/j.jmateco.2010.12.018.

[14]

L. C. Corchón, Comparative statics for aggregative games – The strong concavity case, Math. Social Sci., 28 (1994), 151-165.  doi: 10.1016/0165-4896(94)90001-9.

[15]

L. C. Corchón, On the allocative effects of rent-seeking, J. Public Econom. Theory, 2 (2002), 483-491.  doi: 10.1111/1097-3923.00047.

[16]

L. C. Corchón, The theory of implementation: What did we learn?, Encyclopedia of Complexity and Systems, Springer, 2009.

[17]

L. C. Corchón and A. Mas-Colell, On the stability of best reply and gradient systems with applications to imperfectly competitive models, Econom. Lett., 51 (1996), 59-65.  doi: 10.1016/0165-1765(95)00752-0.

[18]

L. C. Corchón and M. Serena, Contest theory: A survey, in Handbook of Game Theory and Industrial Organization, Edward Elgar Publishing, 2017. doi: 10.2139/ssrn.2811686.

[19]

L. C. Corchón and M. Serena, Properties of contests: Constructing contest success functions from best-responses, working paper of the Max Planck Institute for Tax Law and Public Finance, 2016. Available from: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2882537. doi: 10.2139/ssrn.2882537.

[20]

G. Debreu, Excess demand functions, J. Math. Econom., 1 (1974), 15-21.  doi: 10.1016/0304-4068(74)90032-9.

[21]

A. Dixit, Strategic behavior in contests, Amer. Econom. Rev., 77 (1987), 891-898. 

[22]

G. Epstein and S. Nitzan, The politics of randomness, Social Choice and Welfare, 27 (2006), 423-433.  doi: 10.1007/s00355-006-0125-z.

[23]

F. FallucchiA. Mercatanti and J. Niederreiter, Identifying types in contest experiments, Internat. J. Game Theory, 50 (2021), 39-61.  doi: 10.1007/s00182-020-00738-w.

[24]

Q. Fu and Z. Wu, Contests: Theory and topics, in Oxford Research Encyclopedia of Economics and Finance, Oxford University Press, 2019. doi: 10.1093/acrefore/9780190625979.013.440.

[25]

R. L. Fullerton and R. P. McAfee., Auctioning entry into tournaments, J. Political Economy, 107 (1999), 573-605.  doi: 10.1086/250072.

[26]

D. Furth, Anything goes with heterogeneous, but not always with homogeneous oligopoly, J. Econom. Dynam. Control, 33 (2009), 183-203.  doi: 10.1016/j.jedc.2008.04.012.

[27]

R. J. Gary-Bobo, Cournot-Walras and locally consistent equilibria, J. Econom. Theory, 49 (1989), 10-32.  doi: 10.1016/0022-0531(89)90066-5.

[28]

A. GelderD. Kovenock and B. Roberson, All-pay auctions with ties, Econom. Theory, (2019).  doi: 10.1007/s00199-019-01195-7.

[29]

A. Gelder, D. Kovenock and R. M. Sheremata, Behavior in all-pay auctions with ties, working paper, 2015. doi: 10.2139/ssrn.2683568.

[30]

A. L. Hillman and J. G. Riley, Politically contestable rents and transfers, Economics & Politics, 1 (1989), 17-39.  doi: 10.1111/j.1468-0343.1989.tb00003.x.

[31] M. W. HirschS. Smale and R. L. Devaney, Differential Equations, Dynamical Systems, and an Introduction to Chaos, Academic Press, 1974. 
[32]

H. Jia, Contests with the probability of a draw: A stochastic foundation, Economic Record, 88 (2012), 391-406.  doi: 10.1111/j.1475-4932.2012.00822.x.

[33]

R. R. Mantel, On the characterization of aggregate excess demand, J. Econom. Theory, 7 (1974), 348-353.  doi: 10.1016/0022-0531(74)90100-8.

[34]

S. Nitzan, Collective rent dissipation, Econom. J., 101 (1991), 1522-1534.  doi: 10.2307/2234901.

[35]

T. Peeters and S. Szymanski, Vertical restraints in soccer: Financial fair play and the English premier league, working paper, 2012.

[36]

L. Polishchuk and A. Tonis, Endogenous contest success functions: A mechanism design approach, Econom. Theory, 52 (2013), 271-297.  doi: 10.1007/s00199-011-0622-x.

[37]

P. J. Reny, A characterization of rationalizable consumer behavior, Econometrica, 83 (2015), 175-192.  doi: 10.3982/ECTA12345.

[38]

D. G. Saari, The source of some paradoxes from social choice and probability, J. Econom. Theory, 41 (1987), 1-22.  doi: 10.1016/0022-0531(87)90002-0.

[39]

C. P. Simon and L. Blume, Mathematics for Economists, W. W. Norton and Co, New York, London, 1994.

[40]

S. Skaperdas, Contest success functions, Econom. Theory, 7 (1996), 283-290.  doi: 10.1007/BF01213906.

[41]

S. Skaperdas and B. Grofman, Modeling negative campaigning, Amer. Political Sci. Rev., 89 (1995), 49-61.  doi: 10.2307/2083074.

[42]

H. Sonneneschein, Market excess demand functions, Econometrica, 40 (1972), 549-563.  doi: 10.2307/1913184.

[43]

G. Tullock, Efficient rent-seeking, in Towards a Theory of a Rent-Seeking Society, Texas A & M University Press, 1980, 97–112.

[44]

A. Yildizparlak, An application of contest success functions for draws on European soccer, J. Sports Econom., 19 (2018), 1191-1212.  doi: 10.1177/1527002517716973.

show all references

References:
[1]

D. Acemoglu and M. K. Jensen, Aggregate comparative statics, Games Econom. Behav., 81 (2013), 27-49.  doi: 10.1016/j.geb.2013.03.009.

[2]

J. A. Amegashie, A contest success function with a tractable noise parameter, Public Choice, 126 (2006), 135-144.  doi: 10.1007/s11127-006-2461-z.

[3]

A. Arad and S. Penczynski, Multi-dimensional reasoning in competitive resource allocation games: Evidence from intra-team communication, Mimeo, (2020). Available from: https://www.tau.ac.il/aradayal/MD_Reasoning_in_Resource_Allocation_Games.pdf.

[4]

K. H. Baik, Difference-form contest success functions and effort level in contests, European J. Political Economy, 14 (1998), 685-701.  doi: 10.1016/S0176-2680(98)00032-9.

[5]

A. Barge-Gil, E. Huergo, A. López and L. Moreno, Empirical models of firms' R & D, in Handbook of Game Theory and Industrial Organisation, Volume II, Edward Elgar, Chelthenam, UK; Northampton, MA, USA, 2018,475–516. doi: 10.4337/9781788112789.00027.

[6]

C. Beviá and and L. C. Corchón, Relative difference contest success function, Theory and Decision, 78 (2015), 377-398.  doi: 10.1007/s11238-014-9425-4.

[7]

P. R. Blavatskyy, Contest success function with the possibility of a draw: Axiomatization, J. Math. Econom., 46 (2010), 267-276.  doi: 10.1016/j.jmateco.2009.11.012.

[8]

Y.-K. Che and I. Gale, Difference-form contests and the robustness of all-pay auctions, Games Econom. Behav., 30 (2000), 22-43.  doi: 10.1006/game.1998.0709.

[9]

D. J. Clark and C. Riis, Contest success functions: An extension, Econom. Theory, 11 (1998), 201-204.  doi: 10.1007/s001990050184.

[10]

C. Cohen and A. Sela, Contests with ties, B. E. J. Theor. Econ., 7 (2007), 18pp. doi: 10.2202/1935-1704.1398.

[11]

L. Corchón, Theories of Imperfectly Competitive Markets, Springer, Berlin, Heidelberg, 2001. doi: 10.1007/978-3-662-04498-8.

[12]

L. Corchón and M. Dahm, Foundations for contest success functions, Econom. Theory, 43 (2010), 81-98.  doi: 10.1007/s00199-008-0425-x.

[13]

L. Corchón and M. Dahm, Welfare maximizing contest success functions when the planner cannot commit, J. Math. Econom., 47 (2011), 309-317.  doi: 10.1016/j.jmateco.2010.12.018.

[14]

L. C. Corchón, Comparative statics for aggregative games – The strong concavity case, Math. Social Sci., 28 (1994), 151-165.  doi: 10.1016/0165-4896(94)90001-9.

[15]

L. C. Corchón, On the allocative effects of rent-seeking, J. Public Econom. Theory, 2 (2002), 483-491.  doi: 10.1111/1097-3923.00047.

[16]

L. C. Corchón, The theory of implementation: What did we learn?, Encyclopedia of Complexity and Systems, Springer, 2009.

[17]

L. C. Corchón and A. Mas-Colell, On the stability of best reply and gradient systems with applications to imperfectly competitive models, Econom. Lett., 51 (1996), 59-65.  doi: 10.1016/0165-1765(95)00752-0.

[18]

L. C. Corchón and M. Serena, Contest theory: A survey, in Handbook of Game Theory and Industrial Organization, Edward Elgar Publishing, 2017. doi: 10.2139/ssrn.2811686.

[19]

L. C. Corchón and M. Serena, Properties of contests: Constructing contest success functions from best-responses, working paper of the Max Planck Institute for Tax Law and Public Finance, 2016. Available from: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2882537. doi: 10.2139/ssrn.2882537.

[20]

G. Debreu, Excess demand functions, J. Math. Econom., 1 (1974), 15-21.  doi: 10.1016/0304-4068(74)90032-9.

[21]

A. Dixit, Strategic behavior in contests, Amer. Econom. Rev., 77 (1987), 891-898. 

[22]

G. Epstein and S. Nitzan, The politics of randomness, Social Choice and Welfare, 27 (2006), 423-433.  doi: 10.1007/s00355-006-0125-z.

[23]

F. FallucchiA. Mercatanti and J. Niederreiter, Identifying types in contest experiments, Internat. J. Game Theory, 50 (2021), 39-61.  doi: 10.1007/s00182-020-00738-w.

[24]

Q. Fu and Z. Wu, Contests: Theory and topics, in Oxford Research Encyclopedia of Economics and Finance, Oxford University Press, 2019. doi: 10.1093/acrefore/9780190625979.013.440.

[25]

R. L. Fullerton and R. P. McAfee., Auctioning entry into tournaments, J. Political Economy, 107 (1999), 573-605.  doi: 10.1086/250072.

[26]

D. Furth, Anything goes with heterogeneous, but not always with homogeneous oligopoly, J. Econom. Dynam. Control, 33 (2009), 183-203.  doi: 10.1016/j.jedc.2008.04.012.

[27]

R. J. Gary-Bobo, Cournot-Walras and locally consistent equilibria, J. Econom. Theory, 49 (1989), 10-32.  doi: 10.1016/0022-0531(89)90066-5.

[28]

A. GelderD. Kovenock and B. Roberson, All-pay auctions with ties, Econom. Theory, (2019).  doi: 10.1007/s00199-019-01195-7.

[29]

A. Gelder, D. Kovenock and R. M. Sheremata, Behavior in all-pay auctions with ties, working paper, 2015. doi: 10.2139/ssrn.2683568.

[30]

A. L. Hillman and J. G. Riley, Politically contestable rents and transfers, Economics & Politics, 1 (1989), 17-39.  doi: 10.1111/j.1468-0343.1989.tb00003.x.

[31] M. W. HirschS. Smale and R. L. Devaney, Differential Equations, Dynamical Systems, and an Introduction to Chaos, Academic Press, 1974. 
[32]

H. Jia, Contests with the probability of a draw: A stochastic foundation, Economic Record, 88 (2012), 391-406.  doi: 10.1111/j.1475-4932.2012.00822.x.

[33]

R. R. Mantel, On the characterization of aggregate excess demand, J. Econom. Theory, 7 (1974), 348-353.  doi: 10.1016/0022-0531(74)90100-8.

[34]

S. Nitzan, Collective rent dissipation, Econom. J., 101 (1991), 1522-1534.  doi: 10.2307/2234901.

[35]

T. Peeters and S. Szymanski, Vertical restraints in soccer: Financial fair play and the English premier league, working paper, 2012.

[36]

L. Polishchuk and A. Tonis, Endogenous contest success functions: A mechanism design approach, Econom. Theory, 52 (2013), 271-297.  doi: 10.1007/s00199-011-0622-x.

[37]

P. J. Reny, A characterization of rationalizable consumer behavior, Econometrica, 83 (2015), 175-192.  doi: 10.3982/ECTA12345.

[38]

D. G. Saari, The source of some paradoxes from social choice and probability, J. Econom. Theory, 41 (1987), 1-22.  doi: 10.1016/0022-0531(87)90002-0.

[39]

C. P. Simon and L. Blume, Mathematics for Economists, W. W. Norton and Co, New York, London, 1994.

[40]

S. Skaperdas, Contest success functions, Econom. Theory, 7 (1996), 283-290.  doi: 10.1007/BF01213906.

[41]

S. Skaperdas and B. Grofman, Modeling negative campaigning, Amer. Political Sci. Rev., 89 (1995), 49-61.  doi: 10.2307/2083074.

[42]

H. Sonneneschein, Market excess demand functions, Econometrica, 40 (1972), 549-563.  doi: 10.2307/1913184.

[43]

G. Tullock, Efficient rent-seeking, in Towards a Theory of a Rent-Seeking Society, Texas A & M University Press, 1980, 97–112.

[44]

A. Yildizparlak, An application of contest success functions for draws on European soccer, J. Sports Econom., 19 (2018), 1191-1212.  doi: 10.1177/1527002517716973.

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