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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 and 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.

[2]

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

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

[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.

[6]

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

[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.

[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.

[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.

[10]

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

[11]

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

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

[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.

[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.

[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.

[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.

[18]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[28]

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

[29]

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

[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.

[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.

[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.

[33]

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

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

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

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

[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. 

[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.

[40]

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

[41]

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

[42]

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

[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.

show all references

References:
[1]

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

[2]

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

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

[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.

[6]

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

[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.

[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.

[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.

[10]

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

[11]

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

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

[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.

[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.

[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.

[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.

[18]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[28]

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

[29]

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

[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.

[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.

[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.

[33]

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

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

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

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

[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. 

[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.

[40]

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

[41]

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

[42]

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

[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.

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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