Global dynamics in adaptive models of collective choice with social influence Gian-Italo Bischi

1. Global dynamics in adaptive models of collective choice with social influence Gian-Italo Bischi Università di Urbino gian.bischi@uniurb.it Ugo Merlone Università di Torino merlone@econ.unito.it

2. OUTLINE Discrete-time models that describe the trade-off between repeated individual choices and long run collective behavior • A one-dimensional model that describes binary choice games with externalities. • Thomas Schelling (1978) Micromotives and Macrobehavior, W. W. Norton • Peyton Young (1998) “Individual strategy and Social Structure”, Princeton Univ. • - Akira Namatame “Adaptation and Evolution in Collective Systems”, WS, 2006.

3. Binary choices with externalities Schelling (1973) "Hockey Helmets, Concealed Weapons, and Daylight Saving“ Journal of Conflict Resolution • Should I wear the helmet or not during the hockey match? It depends if the other guys do or not. • Should I carry a weapon or going unarmed? It depends on what other guys do (apply to nations) • Should I take the car or the train ? • Should I invest in R&D or not? (consider spillover effects) • Join or not? (switch watches to daylight saving time or stay on standard time) • Should I dress elegant or not at the annual meeting of my society? • Let’s go to this restaurant, or not? • Should I get annual flu vaccination or not ? • Should I spray the insecticide in my garden or not? • Should I go to vote for my favourite party or not?

4. Modeling social interactions with externalities key simplifications proposed by Schelling: (a) each player has a purely binary choice (e.g. S1 or S2; R or L) (b) the interaction is impersonal, i.e. each player's payoff depends only on the numbers of others making the one choice or the other, not on the identities of these choosers. More formally: Let’s consider a population of n players Let x  [0,1] be the fraction of players that choose strategy R x = 0 means all choose L, x = 1 means all choose R Payoffs are functions R(x) and L(x) defined in [0,1] Each player decides by comparing payoff functions

5. Left choice is preferred on the left, Right choice on the right Should I take (visible) weapons or not? (Coordination games, bandwagon) dotted line: total payoff xR(x) + (1x)L(x) (Dispersion games, snob attitude) Should I take the car or not?

6. Multiperson Prisoner Dilemma: R(x) < L(x) x, with R(1) > L(0) R = cooperate (dominated strategy  x) L = defect (dominant strategy  x ) All choosing R dominates on all choosing L Unique equilibrium: x = 0 [ being R(0) < L(0) ] Globally stable [ being R(x) < L (x) x ] k : R(x) > L(0) for x > k , minimum size of any coalition that can gain by making the dominated choice

7. S1: car tAB = LAB/Vav where Vav= Vmax (1 − n/N) hence U1(x) = S2: train tAB = LAB/VT hence U2(x) = Collective efficiency: xU1(x)+ (1-x)U2(x) = Collective optimum for x = < individual optimum (Nash equilibrium) Car of train? A B U = 1/tAB Population of N agents, n [0,N] choose car. Let x = n/N [0,1] U1 U2 x 1 0

8. Research investments or just spillovers? S1: invest S2: just spillovers U1 = ax – c U2 = bx hopefully b<a b < a - c b > a - c U1 a-c U2 b b U2 a-c U1 x 0 1 x 0 1 -c -c Collective efficiency: xU1 + (1-x)U2 = x(ax-c) +(1-x)bx = (a-b)x2 + (b-c)x Collective optimum for x = 1

9. Compound social games all other agents (collective choice) S1 x S2 1-x All identical individuals (symmetric game) Global interaction (with all agents with the same weight) one agent (individual choice) a,a b,c S1 S2 c,b d,d U1 a U2 U1 = ax + b(1-x) = (a-b)x + b U2 = cx + d(1-x) = (c-d)x + d c d b x 0 1 Collective efficiency: xU1 + (1-x)U2 = (a+d-c-b)x2 + (b+c-2d)x +d

10. Dilemma games: c > a > d > b all other agents (collective choice) S1 x S2 1-x one agent (individual choice) a,a b,c S1 S2 c,b d,d U2 c U1 a U1 = ax + b(1-x) = (a-b)x + b U2 = cx + d(1-x) = (c-d)x + d d b x 0 1

11. Coordination games: a > c and d >b “same strategy” dominates on “different strategy” (imitative, bandwagon) all other agents (collective choice) S1 x S2 1-x one agent (individual choice) a,a b,c S1 S2 c,b d,d U1 a c U2 U1 = ax + b(1-x) = (a-b)x + b U2 = cx + d(1-x) = (c-d)x + d d b 0 x 1

12. Dispersion games: c > a and b > d (reversed inequality wrt coordination) “different strategy” dominates on “same strategy” (snob, not imitative) all other agents (collective choice) S1 x S2 1-x one agent (individual choice) a,a b,c S1 S2 c,b d,d b c U2 U1 = ax + b(1-x) = (a-b)x + b U2 = cx + d(1-x) = (c-d)x + d a U1 d x 0 1

13. A dynamic adjustment is implicitly assumed by Schelling (1973) based on the following assumptions: (i) x will increase whenever R(x) > L(x), decrease when the opposite inequality holds; (ii) interior equilibria x = x* located where : R (x*) = L(x*) (interior equilibria) or (boundary equilibria) x* = 0 provided that R(0) < L(0) x* = 1 provided that R(1) > L(1). No dynamic models are explicitly proposed by Schelling

14. Towards an explicit dynamic model Bischi, Merlone "Global Dynamics in Binary Choice Models with Social Influence“ Journal of Mathematical Sociology, 33:277–302, 2009 Assume discrete time adjustment. In economic and social systems changes over time are usually related to decisions that cannot be continuously revised. If one takes a decision at a given time, very rarely such decision can be modified after an infinitesimal time Time scale is not specified in Schelling (1973, 1978) Schelling (1978, ch.3) describes several situations where individuals make repeated binary choices, with an evident discrete time scale. "The phenomenon of overshooting is a familiar one at the level of individual ...“ consequently … "Numerous social phenomena display cyclical behavior".

15. At time t, xt players are playing strategy R If R (xt ) > L (xt) a fraction of the (1xt) players playing L switch to strategy R If R (xt ) < L (xt) a fraction of the xt players playing R switch to strategy L dR and dLmaximum values of switching fractions l > 0 switching propensity (or speed of reaction) continuos and increasing function with modulates how the switching depends on the difference between realized payoffs In the numerical examples If the payoff functions are continuous, then the map f is continuous Even if L(x) and R(x) are smooth functions, f is not smooth where R(x) = L(x) The graph of f is contained in the strip bounded by two lines (1  δL) x ≤ f(x) ≤ (1  δR) x + δR.

16. limiting case is obtained for λ→+∞ (impulsive players) the switching rate only depends on the sign of the difference between payoffs no matter how much they differ. f (1R)x+R f R (1L)x L x x 1 1 0 0 x 1 0

17. Multiperson prisoner dilemma

18. Payoff functions with one intersection R x* L R x 0 1 case 1 R(0) < L(0) and R(1) > L(1), coordination games i.e. R is preferred at the right of x∗ , L at the left g(⋅)=(2/p)arctan(⋅) (1-dR)x+dR f x* (1-dL)x 1 x 0

19. case 2 R(0) > L(0) and R(1) < L(1), dispersion games i.e. R is preferred at the left of x∗, L at the right g(⋅)=(2/p)arctan(⋅), dL=dR=0.45 L l= l=60 l=40 l=20 R l=10 x* R l=5 L x 1 0 L(x) = 1.5x, R(x)= 0.25 + 0.5x,

20. Both slopes decrease as l or dL or dR increase, i.e. if the propensity to switch increases. dL= 0.2 dR= 0.8 l = 30 dL= 0.4 dR= 0.5 l = 25 x* x* x0 (b) (a) x x 1 0 1 0 fig. 3

21. l=10 l=20 l=35 f 0 x 1 dL = dR = 0.5 l= 1 g(⋅)=(2/p)arctan(⋅), dL=dR=0.5 x 0 0 l 70

22. Should I spray insepticide in my garden? • Will I go to the fest? • Open a shop or not? See also: Granovetter M., (1978), "Threshold Models of Collective Behavior“ The American Journal of Sociology 83(6) Granovetter M. and R. Soong, (1978), "Threshold Models of Diffusion and Collective Behavior", Journal of Mathematical Sociology 9

23. Payoff functions with two intersections x0 Z0 cMax f Z2 cmin x0 cmin Z1 x 1 0 g(⋅)=(2/p)arctan(⋅), dL=dR=0.5 l=6 f R L x 1 0 L(x)= 0.5x, R(x)= - 8 x2 + 12 x - 4,

24. x0 x0 Z0 x0 cMax Z2 cmin cmin Z1 f f x 1 x 1 0 0 g(⋅)=(2/p)arctan(⋅), dL= dR = 0.5 l=10

25. g(⋅)=(2/p)arctan(⋅), dL=dR= 0.4 l = 40 x0 = 0.95 x0 = 0.91 x0 = 0.8 x x x 0 0 1 1 1 0 Z0 Z2 Z3 f Z1 x 1 0

27. +R (1R) (1L) 1 1dL (1R)x+R f (1L)x x x 1 1 0 0

28. l= x x 1 1 0 0 dL = dR = 0.4 dL= 0.25 dR=0.4

29. (b) (a) 1 x2 x2 TL B(0) x3 x1 TR x1 B(0) 0 0 1 x 1 x

30. Concluding remarks • A simple (didactical) discrete time adaptive dynamic models, based on Schelling’s qualitative assumptions, has been proposed to mimic the time evolution of social systems where repeated individual choices lead to the emergence of long-run collective behaviours • These explicit dynamic models allowed us to investigate some global dynamical properties, in particular the structure of the basins of attraction when several attractors coexist. This shows how an adaptive adjustment can be used as an equilibrium selection device that indicates path-dependent evolutions through which different collective long run behaviours can emerge from repeated - step by step - adaptive - myopic individual decisions. • By tuning the values of the parameters we observe some intuitive and expected results as well as some counterintuitive one, mainly related to overshooting effects associated with the peculiar properties of noninvertible iterated maps. • The presence of overshooting should not be seen as an artificial effect or a distortion of reality due to discrete time scale we have considered. Instead, as stressed by Schelling (1978), overshooting and over-reaction arise quite naturally in social systems, due to emotional attitude, impulsivity or lack of information.