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By Lindsey Hock

Thin is In: A Game Theoretic Analysis of the Trend of Unnaturally Thin Models, Its Consequences and Possible Future. By Lindsey Hock. Description.

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By Lindsey Hock

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  1. Thin is In: A Game Theoretic Analysis of the Trend of Unnaturally Thin Models, Its Consequences and Possible Future By Lindsey Hock

  2. 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.

  3. 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

  4. A.2 The relationship of α and β • α is constant but β is a functions of wm • Figure A.2.1

  5. β′(w)<0, β″(w)>0 • Figure A.2.2

  6. A.3 α, β and Stable Equilibrium

  7. 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]

  8. Graphical implication of equation 9

  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.

  10. 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]

  11. C.4 The Payoff Matrix

  12. 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.”

  13. 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*

  14. 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]

  15. They create output “weight.” • The cost of “weight” is c(w), a decreasing function in weight. c(w)=K-w [11]

  16. 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=-1 2w=C+2 W=(C+2)/2 [16] c′(w)=M-2w=-2 2w=C+1 W=(C+1)/2 [15]

  17. 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]

  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.

  19. 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]

  20. Future of the Trend? • F.2 Effect of decreasing Π (holding C constant) • Lim (Π0) p = 1- (C/0+) [using equation 22] • Lim (Π0) p = 1- ∞ • Lim (Π0) 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(C0)p=1 **Fate may rest on changes in culture, social structure, financial cost and accessibility to medical procedures**

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