Social Class and Luck: Some Lessons from Gambler's Ruin and Branching Processes

Political Studies

Volumn 45, Issue 1, 1997, Pages 66-77

  Coram, Alexander T ^a  

^a NONE

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EID: 0345909562     PISSN: 00323217     EISSN: None     Source Type: Journal    
DOI: 10.1111/1467-9248.00070     Document Type: Article

References (24)

1. R. Goodin, Reasons for Welfare (Princeton NJ, Princeton University Press, 1988), p. 294 fn. 27.
2. Samuelson calls this the popular view. P. Samuelson, 'The √N Law and Repeated Risk Taking' in T. Anderson et al. (eds), Probability, Statistics and Mathematics (San Diego CA, Academic, 1989). The fifteen or so colleagues I asked also held this view.
3. Samuelson, 'The √N Law'.
4. Samuelson sees this as a confirmation of Schumpeter's hypothesis that capitalism is like a hotel in which the occupants of the floors are always changing. The √N Law', p. 305.
5. The point can be expressed with more authority, but slightly illogically, as follows. 'For unto everyone that hath shall be given ... but from him that hath not shall be taken away even that which he hath'. Matthew 25: 29.
6. M. Friedman, 'Choice, chance and the personal distribution of income', Journal of Political Economy, 58, (1953) 277-90, p. 290.
7. All figures from E. Wolff, 'Trends in household wealth in the United States, 1962-63 and 1983-89', Review of Income and Wealth, 40, No. 2, (1994), 143-74, p. 153.
8. Samuelson, 'The √N Law', p. 304.
9. More accurately, let Sn be the sum of a sequence of gambles and m the sum of the expectations. Then [| Sn - m |]/n → 0 with probability 1.
10. The law of large numbers actually says the divergence of a from the expected outcome will grow in the same manner as √n. See J. Galambos, Introductory Probability Theory (New York NY, Marcel Dekker, 1984) pp. 191-4, and K. Chung, Elementary Probability Theory with Stochastic Processes (New York NY, Springer, 1975), p. 229 and Samuelson 'The √N Law', p. 292, for a discussion of this point.
11. The law of large numbers actually says the divergence of a from the expected outcome will grow in the same manner as √n. See J. Galambos, Introductory Probability Theory (New York NY, Marcel Dekker, 1984) pp. 191-4, and K. Chung, Elementary Probability Theory with Stochastic Processes (New York NY, Springer, 1975), p. 229 and Samuelson 'The √N Law', p. 292, for a discussion of this point.
12. The law of large numbers actually says the divergence of a from the expected outcome will grow in the same manner as √n. See J. Galambos, Introductory Probability Theory (New York NY, Marcel Dekker, 1984) pp. 191-4, and K. Chung, Elementary Probability Theory with Stochastic Processes (New York NY, Springer, 1975), p. 229 and Samuelson 'The √N Law', p. 292, for a discussion of this point.
13. This problem is covered in the mathematical literature under the generic title of gambler's ruin.
14. Such considerations led Keynes to suggest that 'poor men should not gamble and millionaires should do nothing else'. It follows that 'millionaires are often fortunate fools that have thriven on unfortunate ones'. J. Keynes, A Treatise on Probability (London, Macmillan, 1921), p. 320.
15. M. Farrell, 'Some elementary selection processes in economics', Review of Economic Studies, 37 (1970), 305-19, argues that such a model provides the most appropriate means of analysing outcomes under conditions of uncertainty.
16. k.
17. S. Karlin and M. Taylor, A First Course in Stochastic Processes (New York NY, Academic, 1975), p. 399.
18. Karlin, A First Course, p. 418.
19. See, for example, D. Williams, Probability with Martingales (Cambridge, Cambridge University Press, 1991) pp. 10-3. The probability generating functions are φ(s) = p(1 - qs) and
20. Williams, Probability, p. 10.
21. n(s).
22. This is called Chebyshev's inequality.
23. M. Woodroofe, Probability with Applications (New York NY, McGraw Hill, 1975), p. 241.
24. k where f(s) is the generating function for one unit given in Political Studies (1997), XLV, 78-92