Thin is In: A Game Theoretic Analysis of the Trend of

Unnaturally Thin Models, Its Consequences and Possible

Future

By Lindsey Hock

Description• The concepts of beauty and ideal weight are dynamic in society; it is

obvious that in recent years, the advertising industry has promoted rail-thin models who exhibit a body-type far from the idyllic voluptuous figure of the mid-twentieth century.

• This level of “thinness” extends beyond a promotion of fitness and has been taken to such a degree that it can rarely be achieved by natural means.

• This effect is not only injurious to models (forced to incur the emotional, health-related, or financial costs of maintaining a low-body weight) but also consumers, who may incur similar costs by attempting to achieve the “ideal” or suffer the emotional costs of not being able to achieve it.

• I ask, then, can society do better? • Attempts have been made to alleviate this problem within the

modeling industry, but a game-theoretic analysis will demonstrate the difficulty in achieving a long-term positive outcome as well as provide insight as to the possible future of this trend.

Appendix A: Two Brands Simultaneously Determine the Weight of Their Models

A.1 The First Order Condition for Profit-Maximization• Firms receive profit according to the level of their brand’s “Esteem” which

they are able to generate among consumers• E = level of esteem • Π = Profit• Π = f(E)• Cost of E = c(E)• Π = E – c(E) [1]• E = α(wc–wm) [2]• C(E) = -βwm [3]• Π = α(wc–wm) -(-βwm) [from equations 1,2,3]• Π = α(wc–wm) + βwm [4]• First order condition: dΠ/dw = -αwm+βwm=0 [using equation 4]• αwm=βwm• α=β

• A.2 The relationship of α and β

• α is constant but β is a functions of wm

• Figure A.2.1

• β′(w)<0, β″(w)>0

• Figure A.2.2

A.3 α, β and Stable Equilibrium

Appendix B: The Preferences of Society

• B.1. Utility of Consumers– Uc=f(wm-wc)– (∂Uc/∂wm)>0– Uc=λ(wm-wc) [5]

• B.2. Utility to Brands• Ub=f(Π)= Π [6]• B.3. Welfare Maximization• W=Uc+Ub• W= λ(wm-wc) + Π [from equations 5 and 6] [7]• B.4. Welfare Maximization• W = λ(wm-wc)+α(wc – wm)+ βwm [from equations 4 and 7] [8]• First order condition:• (∂W/∂wm)= λ-α+β=0 [using equation 8]• α= β+λ [9]

• Graphical implication of equation 9

Appendix C: Decision of Cities - Following a Set Limit Results in A

Prisoner’s Dilemma

• C.1. Set-up • 2 players are 2 cities• Each can either cooperate (C) by setting a limit

or defect (D) by not setting a limit• C.2 Payoffs• If a city unilaterally defects, they will take all

profits (Π(high))• If they cities employ the same strategy, they will

split the profit.• If both cooperate, they will receive [Π(high)]/2.• If both defect, they will receive [Π(low)]/2.

• C.3. Calculating [Π(high)] and [Π(low)]• According to Assumption 3, profit to the city is proportional to profit of brands. In this case the

proportion coefficient will be μ.• Πc= μΠb. [10]• Πc(low)= μ[α(wc – wm*′)+ βwm*’] [from equations 4 and 10]• Πc(low)= μ[αwc - αwm*′+ βwm*′]• Πc(low)= μ[αwc + (-α+β)wm*′] [11]• α= β+λ [equation 9, welfare-maximizing condition]• λ= α- β• -λ= -α+β [12]• Πc(low)= μ[αwc + (-λ)wm*′] [from equations 11 and 12] [13]• By a similar process,• Πc(high)= μ[αwc + (-λ)wm*] [14]• Note: Πc(low) is necessarily lower than Πc(high) since wm*′> wm* as explained in Appendix one.• [Πc(low)/2]= (μ/2)[αwc + (-λ)wm*′] [15]• [Πc(high)/2]= (μ/2)[αwc + (-λ)wm*] [16]

C.4 The Payoff Matrix

• C.5. A consideration of the move from {(D,D)} to {(C,C)}• So long as the city maximizes profits with a consideration of welfare, moving to

{(C,C)} under this payoff structure would not be optimum. • However, as demonstrated in Appendix B, a move to {(C,C)} would maximize welfare.

If cities set the limit we would observe wm*′, the optimum from Appendix B.• The necessary condition for this move to take place, would be that Πc(high) must

equal [Πc(low)/2], since if this were true there would be no incentive to defect.• [Πc(low)/2]= Πc(high) [17]• (μ/2)[αwc + (-λ)wm*′]= (μ)[αwc + (-λ)wm*] [from 14,15,17]• (1/2)αwc-(λ/2) wm*′= αwc-λwm*• λwm*-(λ/2) wm*′=(αwc/2)• λ(wm*-.5wm*′)=(αwc/2)• (wm*-.5wm*′)=[(αwc/2)/ λ] [18]• For indifference between C and D, the relationship expressed in equation [18], which

exhibits the relationship between wm* and wm*′ must hold.• The equation thus states that a degree of the difference between these two

parameters depends on the level of λ, which recall parameterizes the degree to which consumers incur a cost associated with their weight straying from the “ideal.”

• C.6. Discussion of λ (=α-β) and β• (wm*-.5wm*′)=[(αwc/2)/(α-β)] [substituting into equation] [19]• Lim(β∞) (wm*-.5wm*′)=0• wm*=.5wm*′• wm*′=2wm*

Appendix D. The Market For Models

• D.1. The market for modelsModels create output “weight” and face demand from brands.Brand’s demand function:

Equation for inverse demand: p=M-w [10]

• They create output “weight.”

• The cost of “weight” is c(w), a decreasing function in weight.

c(w)=K-w [11]

• Profit to the model• Π = Revenue – costs• Π = pw-c(w) [12]• Π = (M-w)w-c(w) [from 10,11,12]• Π = Mw- w2-c(w) [13]• First-order Condition [using 13]• (∂Π /∂w)=M-2w-c′(w)=0• c′(w)=M-2w [14]

• Implications of first order condition (cost structure may vary from country to country and thus c′(w) may vary)

c′(w)=M-2w=-22w=C+1W=(C+1)/2 [15]

c′(w)=M-2w=-12w=C+2W=(C+2)/2 [16]

• Appendix E. Decision-making of models in a one-shot interaction

• 2 players are 2 models• Each has the strategies “unnatural” (U) in which they maintain a

lower weight by unhealthy or unnatural means or “natural” (N) in which they do not.

• Assumptions: • The brand will hire only one of the two models.• If they employ different strategies, the thinner model will get the job

and receive all profit, while the other receives a payoff of 0. • If they employ the same strategy, each has a 50% chance of being

hired. • Payoffs:

– EP(N|N) = EP(high) = .5(Π)+.5(0) = .5Π [17]– EP(U|U) = EP(low) = .5(Π-C)+.5(C) = .5Π-C [18]

Appendix F. Evolutionarily Stable Strategies in the modeling

industry• 2 species of models in the industry: those who

are prone (P) to being swayed by the pressures put on them by the industry and those who are not (N).

• F.1. Formation of the Payoff Matrix• Payoffs = advantage gained (Π) – costs (C)• No advantage exists if both players are of the

same “species”• Species P incurs a cost (C) but species N does

not.

• Mixed Evolutionarily Stable Strategies• Solving for p*:• EP(P|p) = -Cp+( Π-C)(1-p)

[19]• EP(N|p)= 0

[20]• EP(P|p)= EP(N|p)

[21]• -Cp+( Π-C)(1-p)=0 (from equations 19,20,21)• Cp=( Π-C)(1-p)• Cp= Π-pΠ-C+pC• pΠ=Π-C• p = (Π-C)/ Π

• p=1-(C/ Π) [22]

Future of the Trend?• F.2 Effect of decreasing Π (holding C constant)• Lim (Π0) p = 1- (C/0+) [using equation 22]• Lim (Π0) p = 1- ∞• Lim (Π0) p = 0 (since p is bounded below by 0)• F.3 Effect of changing C (holding Π constant)• p=1-(C/ Π) [equation 22]• (∂p/∂C) = (-1/ Π)• As cost increases, p declines. The lower the value of Π,

the faster this decline occurs.• As cost decreases, p increases.• Lim(C0)p=1

**Fate may rest on changes in culture, social structure, financial cost and accessibility to medical procedures**