American Institute of Mathematical Sciences

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October  2020, 7(4): 269-289. doi: 10.3934/jdg.2020019

Approachability in population games

Dario Bauso 1,2,, and  Thomas W. L. Norman 3,

1. 

Jan C. Willems Center for Systems and Control, ENTEG, Fac. Science and Engineering University of Groningen
2. 

Dip. di Ingengneria, Università di Palermo, IT
3. 

Magdalen College, Oxford, UK

* Corresponding author: Dario Bauso

Received  February 2019 Published  October 2020 Early access  July 2020

This paper reframes approachability theory within the context of population games. Thus, whilst a player still aims at driving her average payoff to a predefined set, her opponent is no longer malevolent but instead is extracted randomly at each instant of time from a population of individuals choosing actions in a similar manner. First, we define the notion of 1st-moment approachability, a weakening of Blackwell's approachability. Second, since the endogenous evolution of the population's play is then important, we develop a model of two coupled partial differential equations (PDEs) in the spirit of mean-field game theory: one describing the best-response of every player given the population distribution, the other capturing the macroscopic evolution of average payoffs if every player plays her best response. Third, we provide a detailed analysis of existence, nonuniqueness, and stability of equilibria (fixed points of the two PDEs). Fourth, we apply the model to regret-based dynamics, and use it to establish convergence to Bayesian equilibrium under incomplete information.

Keywords: Approachability, population games, mean-field games.
Mathematics Subject Classification: Primary: 91A16, 91A22; Secondary: 35Q89.
Citation: Dario Bauso, Thomas W. L. Norman. Approachability in population games. Journal of Dynamics & Games, 2020, 7 (4) : 269-289. doi: 10.3934/jdg.2020019
References:
[1]

R. J. Aumann, Utility theory without the completeness axiom, Econometrica, 30 (1962), 445-462.  doi: 10.2307/1913746.  Google Scholar

[2]

R. J. Aumann, Markets with a continuum of traders, Econometrica, 32 (1964), 39-50.  doi: 10.2307/1913732.  Google Scholar

[3] R. J. Aumann and M. B. Maschler, Repeated Games with Incomplete Information, MIT Press, 1995.   Google Scholar
[4]

F. Bagagiolo and D. Bauso, Objective function design for robust optimality of linear control under state-constraints and uncertainty, ESAIM: Control, Optimisation and Calculus of Variations, 17 (2011), 155-177.  doi: 10.1051/cocv/2009040.  Google Scholar

[5]

M. Bardi, Explicit solutions of some linear-quadratic mean field games, Network and Heterogeneous Media, 7 (2012), 243-261.  doi: 10.3934/nhm.2012.7.243.  Google Scholar

[6]

J. Battelle, The Search, Nicholas Brealey Publishing, 2006. Google Scholar

[7]

D. Bauso, E. Lehrer, E. Solan and X. Venel, Attainability in repeated games with vector payoffs, INFORMS Mathematics of Operations Research, 40 (2015), 739-755.  doi: 10.1287/moor.2014.0693.  Google Scholar

[8]

M. Benaïm, J. Hofbauer and S. Sorin, Stochastic approximations and differential inclusions, SIAM Journal on Control and Optimization, 44 (2005), 338–348. doi: 10.1137/S0363012904439301.  Google Scholar

[9]

D. Blackwell, An analog of the minimax theorem for vector payoffs, Pacific J. Math., 6 (1956), 1-8.  doi: 10.2140/pjm.1956.6.1.  Google Scholar

[10]

F. Blanchini, Set invariance in control – a survey, Automatica, 35 (1999), 1747-1768.  doi: 10.1016/S0005-1098(99)00113-2.  Google Scholar

[11]

L. E. Blume, A. Brandenburger and E. Dekel, Lexicographic probabilities and choice under uncertainty, Econometrica, 59 (1991), 61-79.  doi: 10.2307/2938240.  Google Scholar

[12] N. Cesa-Bianchi and G. Lugosi, Prediction, Learning and Games, Cambridge University Press, 2006.  doi: 10.1017/CBO9780511546921.  Google Scholar
[13]

B. Edelman, M. Ostrovsky and M. Schwarz, Internet advertising and the generalised second-price auction: Selling billions of dollars worth of keywords, American Economic Review, 97 (2007), 242-259.   Google Scholar

[14]

N. J. Elliot and N. J. Kalton, The existence of value in differential games of pursuit and evasion, J. Differential Equations, 12 (1972), 504-523.  doi: 10.1016/0022-0396(72)90022-8.  Google Scholar

[15]

Jeffrey C. Ely and William H. Sandholm, Evolution in Bayesian games I: Theory, Games and Economic Behavior, 53 (2005), 83-109.  doi: 10.1016/j.geb.2004.09.003.  Google Scholar

[16]

D. Foster and R. Vohra, Regret in the on-line decision problem, Games and Economic Behavior, 29 (1999), 7-35.  doi: 10.1006/game.1999.0740.  Google Scholar

[17]

John C. Harsanyi, Games with incomplete information played by 'bayesian' players, i–iii. part ii. Bayesian equilibrium points, Management Science, 14 (1968), 320-334.  doi: 10.1287/mnsc.14.5.320.  Google Scholar

[18]

S. Hart, Adaptive heuristics, Econometrica, 73 (2005), 1401-1430.  doi: 10.1111/j.1468-0262.2005.00625.x.  Google Scholar

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S. Hart and A. Mas-Colell, A general class of adaptive strategies, Journal of Economic Theory, 98 (2001), 26-54.  doi: 10.1006/jeth.2000.2746.  Google Scholar

[20]

S. Hart and A. Mas-Colell, Regret-based continuous-time dynamics, Games and Economic Behavior, 45 (2003), 375-394.  doi: 10.1016/S0899-8256(03)00178-7.  Google Scholar

[21]

M. Y. Huang, P. E. Caines and R. P. Malhamé, Large population stochastic dynamic games: Closed loop McKean-Vlasov systems and the nash certainty equivalence principle, Communications in Information and Systems, 6 (2006), 221-252.  doi: 10.4310/CIS.2006.v6.n3.a5.  Google Scholar

[22]

M. Y. Huang, P. E. Caines and R. P. Malhamé, Large-population cost-coupled LQG problems with non-uniform agents: Individual-mass behaviour and decentralized $\epsilon$-nash equilibria, IEEE Trans. on Automatic Control, 9 (2007), 1560-1571.  doi: 10.1109/TAC.2007.904450.  Google Scholar

[23]

M. Y. Huang, P. E. Caines and R. P. Malhamé, Individual and mass behaviour in large population stochastic wireless power control problems: Centralized and nash equilibrium solutions., In Proc. of the IEEE Conference on Decision and Control, volume 42, pages 98–103, HI, USA, December 2003. Google Scholar

[24]

B. Jovanovic and R. W. Rosenthal, Anonymous sequential games, Journal of Mathematical Economics, 17 (1988), 77-87.  doi: 10.1016/0304-4068(88)90029-8.  Google Scholar

[25]

J.-M. Lasry and P.-L. Lions, Jeux à champ moyen. I. Le cas stationnaire, Comptes Rendus Mathematique, 343 (2006), 619-625.  doi: 10.1016/j.crma.2006.09.019.  Google Scholar

[26]

J.-M. Lasry and P.-L. Lions, Jeux à champ moyen. II. horizon fini et controle optimal, Comptes Rendus Mathematique, 343 (2006), 679-684.  doi: 10.1016/j.crma.2006.09.018.  Google Scholar

[27]

J.-M. Lasry and P.-L. Lions, Mean field games, Japanese Journal of Mathematics, 2 (2007), 229-260.  doi: 10.1007/s11537-007-0657-8.  Google Scholar

[28]

E. Lehrer, Allocation processes in cooperative games, International Journal of Game Theory, 31 (2002), 341-351.  doi: 10.1007/s001820200123.  Google Scholar

[29]

E. Lehrer, Approachability in infinite dimensional spaces, International Journal of game Theory, 31 (2002), 253-268.  doi: 10.1007/s001820200115.  Google Scholar

[30]

E. Lehrer, A wide range no-regret theorem, Games and Economic Behavior, 42 (2003), 101-115.  doi: 10.1016/S0899-8256(03)00032-0.  Google Scholar

[31]

E. Lehrer and E. Solan, Excludability and bounded computational capacity strategies, Mathematics of Operations Research, 31 (2006), 637-648.  doi: 10.1287/moor.1060.0211.  Google Scholar

[32]

E. Lehrer, E. Solan and D. Bauso, Repeated games over networks with vector payoffs: the notion of attainability, in Proceedings of the NetGCoop 2011, IEEE, Paris, France, 2011. Google Scholar

[33]

E. Lehrer and S. Sorin, Minmax via differential inclusion, Journal of Convex Analysis, 14(2): 271–273, 2007.  Google Scholar

[34] M. Maschler, E. Solan and S. Zamir, Game Theory, Cambridge University Press, Cambridge, 2013.  doi: 10.1017/CBO9780511794216.  Google Scholar
[35]

E. Roxin, Axiomatic approach in differential games, J. Optim. Theory Appl., 3 (1969), 153-163.  doi: 10.1007/BF00929440.  Google Scholar

[36] W. H. Sandholm, Population Games and Evolutionary Dynamics, MIT Press, Cambridge, MA, 2010.   Google Scholar
[37]

W. H. Sandholm, Evolution in Bayesian games II: Stability of purified equilibria, Journal of Economic Theory, 136 (2007), 641-667.  doi: 10.1016/j.jet.2006.10.003.  Google Scholar

[38]

A. S. Soulaimani, M. Quincampoix and S. Sorin, Approachability theory, discriminating domain and differential games, SIAM Journal of Control and Optimization, 48 (2009), 2461-2479.   Google Scholar

[39]

P. Varaiya, The existence of solutions to a differential game, SIAM Journal of Control and Optimization, 5 (1967), 153-162.  doi: 10.1137/0305009.  Google Scholar

[40]

H. R. Varian, Position auctions, International Journal of Industrial Organization, 25 (2007), 1163-1178.  doi: 10.1016/j.ijindorg.2006.10.002.  Google Scholar

[41]

N. Vieille, Weak approachability, Mathematics of Operations Research, 17 (1992), 781-791.  doi: 10.1287/moor.17.4.781.  Google Scholar

[42]

John von Neumann, Zur Theorie der Gesellschaftsspiele, Math. Ann., 100 (1928), 295-320.  doi: 10.1007/BF01448847.  Google Scholar

[43]

S. Zamir, Bayesian games: Games with incomplete information, in Computational Complexity, Vol. 1–6, Springer, New York, 2012. doi: 10.1007/978-1-4614-1800-9_16.  Google Scholar

show all references

References:
[1]

R. J. Aumann, Utility theory without the completeness axiom, Econometrica, 30 (1962), 445-462.  doi: 10.2307/1913746.  Google Scholar

[2]

R. J. Aumann, Markets with a continuum of traders, Econometrica, 32 (1964), 39-50.  doi: 10.2307/1913732.  Google Scholar

[3] R. J. Aumann and M. B. Maschler, Repeated Games with Incomplete Information, MIT Press, 1995.   Google Scholar
[4]

F. Bagagiolo and D. Bauso, Objective function design for robust optimality of linear control under state-constraints and uncertainty, ESAIM: Control, Optimisation and Calculus of Variations, 17 (2011), 155-177.  doi: 10.1051/cocv/2009040.  Google Scholar

[5]

M. Bardi, Explicit solutions of some linear-quadratic mean field games, Network and Heterogeneous Media, 7 (2012), 243-261.  doi: 10.3934/nhm.2012.7.243.  Google Scholar

[6]

J. Battelle, The Search, Nicholas Brealey Publishing, 2006. Google Scholar

[7]

D. Bauso, E. Lehrer, E. Solan and X. Venel, Attainability in repeated games with vector payoffs, INFORMS Mathematics of Operations Research, 40 (2015), 739-755.  doi: 10.1287/moor.2014.0693.  Google Scholar

[8]

M. Benaïm, J. Hofbauer and S. Sorin, Stochastic approximations and differential inclusions, SIAM Journal on Control and Optimization, 44 (2005), 338–348. doi: 10.1137/S0363012904439301.  Google Scholar

[9]

D. Blackwell, An analog of the minimax theorem for vector payoffs, Pacific J. Math., 6 (1956), 1-8.  doi: 10.2140/pjm.1956.6.1.  Google Scholar

[10]

F. Blanchini, Set invariance in control – a survey, Automatica, 35 (1999), 1747-1768.  doi: 10.1016/S0005-1098(99)00113-2.  Google Scholar

[11]

L. E. Blume, A. Brandenburger and E. Dekel, Lexicographic probabilities and choice under uncertainty, Econometrica, 59 (1991), 61-79.  doi: 10.2307/2938240.  Google Scholar

[12] N. Cesa-Bianchi and G. Lugosi, Prediction, Learning and Games, Cambridge University Press, 2006.  doi: 10.1017/CBO9780511546921.  Google Scholar
[13]

B. Edelman, M. Ostrovsky and M. Schwarz, Internet advertising and the generalised second-price auction: Selling billions of dollars worth of keywords, American Economic Review, 97 (2007), 242-259.   Google Scholar

[14]

N. J. Elliot and N. J. Kalton, The existence of value in differential games of pursuit and evasion, J. Differential Equations, 12 (1972), 504-523.  doi: 10.1016/0022-0396(72)90022-8.  Google Scholar

[15]

Jeffrey C. Ely and William H. Sandholm, Evolution in Bayesian games I: Theory, Games and Economic Behavior, 53 (2005), 83-109.  doi: 10.1016/j.geb.2004.09.003.  Google Scholar

[16]

D. Foster and R. Vohra, Regret in the on-line decision problem, Games and Economic Behavior, 29 (1999), 7-35.  doi: 10.1006/game.1999.0740.  Google Scholar

[17]

John C. Harsanyi, Games with incomplete information played by 'bayesian' players, i–iii. part ii. Bayesian equilibrium points, Management Science, 14 (1968), 320-334.  doi: 10.1287/mnsc.14.5.320.  Google Scholar

[18]

S. Hart, Adaptive heuristics, Econometrica, 73 (2005), 1401-1430.  doi: 10.1111/j.1468-0262.2005.00625.x.  Google Scholar

[19]

S. Hart and A. Mas-Colell, A general class of adaptive strategies, Journal of Economic Theory, 98 (2001), 26-54.  doi: 10.1006/jeth.2000.2746.  Google Scholar

[20]

S. Hart and A. Mas-Colell, Regret-based continuous-time dynamics, Games and Economic Behavior, 45 (2003), 375-394.  doi: 10.1016/S0899-8256(03)00178-7.  Google Scholar

[21]

M. Y. Huang, P. E. Caines and R. P. Malhamé, Large population stochastic dynamic games: Closed loop McKean-Vlasov systems and the nash certainty equivalence principle, Communications in Information and Systems, 6 (2006), 221-252.  doi: 10.4310/CIS.2006.v6.n3.a5.  Google Scholar

[22]

M. Y. Huang, P. E. Caines and R. P. Malhamé, Large-population cost-coupled LQG problems with non-uniform agents: Individual-mass behaviour and decentralized $\epsilon$-nash equilibria, IEEE Trans. on Automatic Control, 9 (2007), 1560-1571.  doi: 10.1109/TAC.2007.904450.  Google Scholar

[23]

M. Y. Huang, P. E. Caines and R. P. Malhamé, Individual and mass behaviour in large population stochastic wireless power control problems: Centralized and nash equilibrium solutions., In Proc. of the IEEE Conference on Decision and Control, volume 42, pages 98–103, HI, USA, December 2003. Google Scholar

[24]

B. Jovanovic and R. W. Rosenthal, Anonymous sequential games, Journal of Mathematical Economics, 17 (1988), 77-87.  doi: 10.1016/0304-4068(88)90029-8.  Google Scholar

[25]

J.-M. Lasry and P.-L. Lions, Jeux à champ moyen. I. Le cas stationnaire, Comptes Rendus Mathematique, 343 (2006), 619-625.  doi: 10.1016/j.crma.2006.09.019.  Google Scholar

[26]

J.-M. Lasry and P.-L. Lions, Jeux à champ moyen. II. horizon fini et controle optimal, Comptes Rendus Mathematique, 343 (2006), 679-684.  doi: 10.1016/j.crma.2006.09.018.  Google Scholar

[27]

J.-M. Lasry and P.-L. Lions, Mean field games, Japanese Journal of Mathematics, 2 (2007), 229-260.  doi: 10.1007/s11537-007-0657-8.  Google Scholar

[28]

E. Lehrer, Allocation processes in cooperative games, International Journal of Game Theory, 31 (2002), 341-351.  doi: 10.1007/s001820200123.  Google Scholar

[29]

E. Lehrer, Approachability in infinite dimensional spaces, International Journal of game Theory, 31 (2002), 253-268.  doi: 10.1007/s001820200115.  Google Scholar

[30]

E. Lehrer, A wide range no-regret theorem, Games and Economic Behavior, 42 (2003), 101-115.  doi: 10.1016/S0899-8256(03)00032-0.  Google Scholar

[31]

E. Lehrer and E. Solan, Excludability and bounded computational capacity strategies, Mathematics of Operations Research, 31 (2006), 637-648.  doi: 10.1287/moor.1060.0211.  Google Scholar

[32]

E. Lehrer, E. Solan and D. Bauso, Repeated games over networks with vector payoffs: the notion of attainability, in Proceedings of the NetGCoop 2011, IEEE, Paris, France, 2011. Google Scholar

[33]

E. Lehrer and S. Sorin, Minmax via differential inclusion, Journal of Convex Analysis, 14(2): 271–273, 2007.  Google Scholar

[34] M. Maschler, E. Solan and S. Zamir, Game Theory, Cambridge University Press, Cambridge, 2013.  doi: 10.1017/CBO9780511794216.  Google Scholar
[35]

E. Roxin, Axiomatic approach in differential games, J. Optim. Theory Appl., 3 (1969), 153-163.  doi: 10.1007/BF00929440.  Google Scholar

[36] W. H. Sandholm, Population Games and Evolutionary Dynamics, MIT Press, Cambridge, MA, 2010.   Google Scholar
[37]

W. H. Sandholm, Evolution in Bayesian games II: Stability of purified equilibria, Journal of Economic Theory, 136 (2007), 641-667.  doi: 10.1016/j.jet.2006.10.003.  Google Scholar

[38]

A. S. Soulaimani, M. Quincampoix and S. Sorin, Approachability theory, discriminating domain and differential games, SIAM Journal of Control and Optimization, 48 (2009), 2461-2479.   Google Scholar

[39]

P. Varaiya, The existence of solutions to a differential game, SIAM Journal of Control and Optimization, 5 (1967), 153-162.  doi: 10.1137/0305009.  Google Scholar

[40]

H. R. Varian, Position auctions, International Journal of Industrial Organization, 25 (2007), 1163-1178.  doi: 10.1016/j.ijindorg.2006.10.002.  Google Scholar

[41]

N. Vieille, Weak approachability, Mathematics of Operations Research, 17 (1992), 781-791.  doi: 10.1287/moor.17.4.781.  Google Scholar

[42]

John von Neumann, Zur Theorie der Gesellschaftsspiele, Math. Ann., 100 (1928), 295-320.  doi: 10.1007/BF01448847.  Google Scholar

[43]

S. Zamir, Bayesian games: Games with incomplete information, in Computational Complexity, Vol. 1–6, Springer, New York, 2012. doi: 10.1007/978-1-4614-1800-9_16.  Google Scholar

Figure 1.  Payoff space of Prisoner's dilemma: State space $ X = conv\{(3, 3), (1, 1), (0, 4), (4, 0)\} $ (boundary a solid line), supporting hyperplane $ H $ (dot-dashed line) passing through the barycenter, vector field $ dx(t) $ pointing towards $ (\frac{3}{2}, \frac{7}{2}) $ for those who cooperate (region below $ H $) and towards $ (\frac{5}{2}, \frac{1}{2}) $ for those who defect (region above $ H $), $ conv\{(\frac{3}{2}, \frac{7}{2}), (\frac{5}{2}, \frac{1}{2})\} $ is set of approachable points with population strategy $ q = ((\frac{1}{2}, \frac{1}{2}), (\frac{1}{2}, \frac{1}{2})) $, barycenter is self-confirmed with uniform distribution over $ X $
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Figure 2.  Regret space of the Prisoner's dilemma: State space $ X = conv\{(-1, 0), (0, 1)\} $ (solid line), initial distribution $ \rho(x, 0) $ (grey area), and vector field $ dx(t) $ converging to $ y = (-0.5, 0.5) $
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Figure 3.  Regret space of the coordination game: State space $ X = conv\{(-1, 0), (0, 1), (0, -2), (2, 0)\} $ (boundary a solid line), and vector field $ dx(t) $ converging to $ (1, 0) $ (grey area) and $ (0, -1) $ (white area), approachable point is $ y = (0, -1) $, set of approachable points is $ conv\{(1, 0), (0, -1)\} $ (dashed line) with mixed population strategy $ q = (\frac{2}{3}, \frac{1}{3}) $
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Figure 4.  Regret space of parametric game with $ a< 0 < b $: State space $ X = conv\{(0, a), (-a, 0), (-b, 0), (0, b)\} $ (boundary a solid line), vector field $ dx(t) $ converging to $ (0, a) $ which is also an approachable vertex with population strategy $ q = (1, 0) $, supporting hyperplane $ H $ (dot-dashed line) intersects $ X $ only at one point (the vertex)
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Figure 5.  Regret space of parametric game with $ 0<b < a $: State space $ X = conv\{(0, a), (-a, 0), (-b, 0), (0, b)\} $ (boundary a solid line), supporting hyperplane $ H $ (dot-dashed line) passing through the vertex $ (-b, 0) $, vector field $ dx(t) $ converging to $ (0, b) $ left of $ H $ and to $ (-b, 0) $ right of $ H $, $ conv\{(0, b), (-b, 0)\} $ is set of approachable points with population strategy $ q = (0, 1) $, vertex $ (-b, 0) $ is not self-confirmed, while vertex $ (0, a) $ is self-confirmed with population strategy $ q = (1, 0) $
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