Global dynamics in adaptive modelsof collective choice with social influence

Gian-Italo BischiUniversità di Urbinogian.bischi@uniurb.it

Ugo MerloneUniversità di Torinomerlone@econ.unito.it

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.

OUTLINE

-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.

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?

Binary choices with externalities

Schelling (1973) "Hockey Helmets, Concealed Weapons, and Daylight Saving“ Journal of Conflict Resolution

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 Rx = 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

Should I take (visible) weapons or not ?

Should I take the car or not ?

dotted line:total payoff xR(x) + (1x)L(x)

Left choice is preferred on the left, Right choice on the right

(Coordination games, bandwagon)

(Dispersion games, snob attitude)

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

k : R(x) > L(0) for x > k , minimum size of any coalition that can gain by making the dominated choice

Unique equilibrium: x = 0 [ being R(0) < L(0) ]Globally stable [ being R(x) < L (x) x ]

Car of train? A B U = 1/tAB

S1: car tAB = LAB/Vav where Vav= Vmax (1 − n/N) hence U1(x) = )1(max xL

V

AB

Population of N agents, n [0,N] choose car. Let x = n/N [0,1]

S2: train tAB = LAB/VT hence U2(x) = T

AB

V

L

0 1 x

U2

U1

T

AB

V

L

max

AB

V

L

max

max

TV V

V

max

max2TV V

V

max

max2TV V

V

Collective efficiency: xU1(x)+ (1-x)U2(x) =

Collective optimum for x = < individual optimum (Nash equilibrium)

L

Vx

L

VVx

L

V TT

max2max

Research investments or just spillovers?

S1: invest S2: just spillovers

U1 = ax – c U2 = bx hopefully b<a

0 1 x

U2

U1

b < a - c

-c

a-c

b

0 1 x

U2

U1

b > a - c

-c

a-cb

Collective efficiency: xU1 + (1-x)U2 = x(ax-c) +(1-x)bx = (a-b)x2 + (b-c)x

Collective optimum for x = 1

Compound social games

one agent(individual choice)

all other agents(collective choice)

S1

S2

1-x

S2

S1

x

a,a b,c

c,b d,d

All identical individuals(symmetric game)

Global interaction (with all agents with the same weight)

U1 = ax + b(1-x) = (a-b)x + b

U2 = cx + d(1-x) = (c-d)x + d

0 1 x

U2

U1a

b

d

c

Collective efficiency: xU1 + (1-x)U2 = (a+d-c-b)x2 + (b+c-2d)x +d

one agent(individual choice)

all other agents(collective choice)

S1

S2

1-x

S2

S1

x

a,a b,c

c,b d,d

0 1 x

U2

U1

c

b

d

a

Dilemma games: c > a > d > b

U1 = ax + b(1-x) = (a-b)x + b

U2 = cx + d(1-x) = (c-d)x + d

one agent(individual choice)

all other agents(collective choice)

S1

S2

1-x

S2

S1

x

a,a b,c

c,b d,d

0 1 x

U2

U1a

b

d

c

Coordination games: a > c and d >b “same strategy” dominates on “different strategy” (imitative, bandwagon)

U1 = ax + b(1-x) = (a-b)x + b

U2 = cx + d(1-x) = (c-d)x + d

one agent(individual choice)

all other agents(collective choice)

S1

S2

1-x

S2

S1

x

a,a b,c

c,b d,d

0 1 x

U2

U1

c

d

b

a

Dispersion games: c > a and b > d (reversed inequality wrt coordination)

“different strategy” dominates on “same strategy” (snob, not imitative)

U1 = ax + b(1-x) = (a-b)x + b

U2 = cx + d(1-x) = (c-d)x + d

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

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".

Towards an explicit dynamic model

Bischi, Merlone "Global Dynamics in Binary Choice Models with Social Influence“Journal of Mathematical Sociology, 33:277–302, 2009Assume 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

At time t, xt players are playing strategy RIf R (xt ) > L (xt) a fraction of the (1xt) players playing L switch to strategy RIf R (xt ) < L (xt) a fraction of the xt players playing R switch to strategy L

1

( ) ( ) 1 ( ) ( )( )

( ) ( ) ( ) ( )

t R t t t t t

t t

t L t t t t t

x g R x L x x if R x L xx f x

x g L x R x x if R x L x

switching propensity (or speed of reaction) R and L maximum values of switching fractions

If the payoff functions are continuous, then the map f is continuousEven 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.

1,0: g 1)(0)0( gandgcontinuos and increasing function with

modulates how the switching depends on the difference between realized payoffs

arctan2

gIn the numerical examples

1

(1 ) ( ) ( )

( ) ( ) ( )

(1 ) ( ) ( )

R t R t t

t t t t t

L t t t

x if R x L x

x f x x if R x L x

x if R x L x

the switching rate only depends on the sign of the differencebetween payoffs no matter how much they differ.

limiting case is obtained for λ→+∞ (impulsive players)

10 x 10 x

f

(1 L

)x

(1 R

)x+ R

LR

x 10

*1x

*2xf

xxf L 1)(

)()(1)( xRxLgxxf L

Multiperson prisoner dilemma

Payoff functions with one intersectioncase 1R(0) < L(0) and R(1) > L(1), coordination gamesi.e. R is preferred at the right of x∗ , L at the left

x 10

x*

f

g( )=(2/⋅ )arctan( ) ⋅

(1- L)x

(1-R)x+

L

R

x 10

x*

R

case 2R(0) > L(0) and R(1) < L(1), dispersion gamesi.e. R is preferred at the left of x∗, L at the right

L

R

x 10

x*

L(x) = 1.5x, R(x)= 0.25 + 0.5x,

=10

=5

=20=40

=60

=

g( )=(2/⋅ )arctan( ), ⋅ LR

L

R

Both slopes decrease as or L or R increase, i.e. if the propensity to switch increases.

x 10

x*

x0 (a)

x 10

x*

(b)

LR LR

700

1

0

x

f

=10

x 10

=35=20

g( )=(2/⋅ )arctan( ), ⋅ LR

L R =

• 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 CollectiveBehavior", Journal of Mathematical Sociology 9

x 10

*1x

*2x

f

x0

x0

g( )=(2/⋅ )arctan( ), ⋅ LR =6

L

R

x 10

*1x

*2xf

Payoff functions with two intersections

L(x)= 0.5x, R(x)= - 8 x2 + 12 x - 4,

cmin

cmin

cMax

Z2

Z1

Z0

10

*2x

x0

x0

10

*1x

*2x

ff

cmin

cMax

Z2

Z1

Z0

*1x

cmin

*( 1)1x

*( 1)1x

x x

g( )=(2/⋅ )arctan( ), ⋅ LR =10

*2x

f

*1x

Z2

Z1

Z0

Z3

10 x

*( 1),11x

*( 1),21x

g( )=(2/⋅ )arctan( ), ⋅

LR= 40

*1x

*1x *

1x

10 x 10 x 10 x

x0 = 0.8 x0 = 0.91 x0 = 0.95

10 x 10 x

f

(1 L

)x

(1 R

)x+ R

1

1L

(1L) *2x

*1x

(1R) *2x +R

10 x 10 x

L R

=

LR

x

1

10

0

x1

x2

x3

TL

TR

x 1

x1

x2

(a) (b)

B(0)

B(0)

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.

Concluding remarks