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Posts Tagged ‘Economics’

This describes a coordination failure with farmers being unable to collude because incentives to cheat and defect are high. The risk in coordination lies mostly with a farmer who can see that he is able to cheat, which would allow the farmer who did not cheat to bear the entire risk of his crops being destroyed.

This is similar to the prisoner’s dilemma where both players will miss a Nash Equilibrium and a socially optimal strategy by playing their dominant strategies.

Furthermore it touches upon the notion that reducing corruption and increasing trust are prerequisites for meaningful economic development.

Palanpur farmers sow their winter crops several weeks after the date at which yields would be maximised. The farmers do not doubt that earlier plantings would give them larger harvests, but no one, the farmer explained, is willing to be the first to plant, as the seeds on any lone plot would be quickly eaten by birds. I asked if a larger group of farmers, perhaps relatives, had ever agreed to sow earlier, all planting on the same day to minimise the loses. “If we knew how to do that,” he said looking up from his hoe at me, “we would not be poor.”

Via “Origin of Wealth” (chpt Design Spaces), Eric D. Beinhocker.

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I was rearranging my closet and bookshelf and I dusted off a notebook that I ended up using for from Econ 142. I found a particularly cool problem that I encountered back than, that I will reproduce here.

In the game Former Soviet Union Roulette, a number of bullets are loaded into a revolver with six chambers; an individual then points the revolver at his head, pulls the trigger, and is killed if and only if the revolver goes off. Assume the individual must play this game; that he is an expected-utility maximizer; and that each chamber is equally likely to be in firing position, so if the number of bullets is b his probability of being killed is b/6. Suppose further that the maximum amount he is willing to pay to have one bullet removed from a gun initially containing only one bullet is $x, and the maximum amount he is willing to pay to have one bullet removed from a gun initially containing 4 bullets is $y, where x and y are both finite. Finally, suppose that he prefers more money to less and that he prefers life (even after paying $x or $y) to death. Let UD denote his von Neumann-Morgenstern utility when dead, which is assumed to be independent of how much he paid (as suggested by empirical studies of the demand for money); and let UA0, UAx, and UAy denote his von Neumann-Morgenstern utilities when alive after paying $0, $x, or $y respectively.

  1. What restrictions are placed on UD, UA0, UAx, and UAy by the assumption that he prefers more money to less when alive?
  2. What restrictions are placed on UD, UA0, UAx, and UAy by the assumption that he prefers life (even after paying $x or $y) to death?
  3. Is it possible to tell from the information given above whether x > y for an expected utility maximizer? Does it matter whether he is risk-averse? Explain.

First things first, lets just write down what we know.

  • Expected Utility maximizer.
  • Probably concave utility function where second derivative is negative.
  • Homo Economicus
  • b/6, b= bullets, each additional bullet adds like a ~16% chance of being killed
  • He is willing to pay an amount $x to have a bullet removed from a gun that only has one bullet. Essentially letting him live.
  • He is willing to pay an amount $y to have a bullet removed from a gun that has 4 bullets.
  • More $ > Less $ is his preference.
  • Life > Death is also his preference.
  • UD is independent of payment.

Utility functions defined:

  • UD = Utility of being dead
  • UA0 = Utility of being alive and paying $0
  • UAx = Utility of being alive and paying $x
  • UAy = Utility of being alive and paying $y

Scaling of utilities gives us

  • U(a1) = 0
  • U(a2) = U(a3) = U(a4) = 1/3
  • U(a5) = U(a6) = U(a7) = 2/3
  • U(a8) = 1

1.) Using the last two bits of information allows us to solve the first part.  UA0 > UAy and UAx > UD

2.) UA0, UAy, UAx > UD
3.) For the last question we have

  • EU($x) = UD*(1/6)+EUAx*(5/6)
  • EU(Nothing) = UD*(1/6)+UA0*(5/6)
  • EU(Removing Bullet at Cost $x) = UAx
  • EU(Nothing) = UD*(1/6)+ UA0*(5/6)
  • EU(Removing Bullet at Cost $y) = UD*3/6 + UAy*(3/6)
  • U’ > more money, happier, life over death, UA0 > UAx ~ UAy > UD

Setting up a system of equations for this situation yields us

1/6*UD + 5/6 UA0 = UAx
3/6UD + 3/6*Ay = 4/6*UD + 2/6UA0
+ 3/6*Ay = 1/6*UD + 2/6UA0

Bottom equation falls out since it can be rearranged to look like the second one.

1/6*UD + 5/6 UA0 = UAx
3/6UD + 3/6*Ay = 4/6*UD + 2/6UA0

Solving yields us:

UA0 = 2UAx – UAy
UA0 = UAx   + Uax – UAy (this is positive)
So UAx > UAy

Addendum
This paper attempts to reconcile the Russian Roulette problem to Kahneman and Tversky’s Prospect Theory.

http://lishu.psych.ac.cn/doc/A%20closer%20look%20at%20the%20Russian%20roulette%20problem%202009.pdf

Utilizing the Russian roulette problem as an exemplar, Kahneman and Tversky (1979) developed a weighting function p to explain that the Allais Paradox arises because people behave so as to maximize overall value rather than expected utility (EU). Following the way that ‘‘overweighting of small probabilities” originated from the Russian roulette problem, this research measured individuals’ willingness to pay (WTP) as well as their happiness for a reduction of the probability of death, and examined whether the observed figures were compatible with the nonlinearity of the weighting function. Data analysis revealed that the nonlinear properties estimated by straight measures differed from those derived from preferential choices [D. Kahneman, A. Tversky, Prospect theory: an analysis of decision under risk, Econometrica 47 (1979) 263–291] and formulated by [A. Tversky, D. Kahneman, Advances in prospect theory: Cumulative representation of uncertainty, Journal of Risk and Uncertainty 5 (1992) 297–323]. The controversies and questions to the proposed properties of the decision weight were discussed. An attempt was made to draw the research attention from which function was being maximized to whether people behave as if they were trying to maximize some generalized expectation.

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