Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-10T09:29:22.171Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotics for the time of ruin in the war of attrition

Part of: Stochastic processes Game theory

Published online by Cambridge University Press:  26 June 2017

Philip A. Ernst  and
Ilie Grigorescu
Philip A. Ernst*
Affiliation:
Rice University
Ilie Grigorescu*
Affiliation:
University of Miami
*
* Postal address: Department of Statistics, Rice University, 6100 Main Street, Houston, TX 77005, USA. Email address: philip.ernst@rice.edu
** Postal address: Department of Mathematics, University of Miami, 1365 Memorial Drive, Coral Gables, FL 33146, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider two players, starting with m and n units, respectively. In each round, the winner is decided with probability proportional to each player's fortune, and the opponent loses one unit. We prove an explicit formula for the probability p(m, n) that the first player wins. When m ~ Nx0, n ~ Ny0, we prove the fluid limit as N → ∞. When x0 = y0, z → p(N, N + z√N) converges to the standard normal cumulative distribution function and the difference in fortunes scales diffusively. The exact limit of the time of ruin τN is established as (T - τN) ~ N-βW1/β, β = ¼, T = x0 + y0. Modulo a constant, W ~ χ21(z02 / T2).

Keywords

War of attritiongambler's ruinevolutionary game theorydiffusive scalingnoncentered chi-squared

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 91A60: Probabilistic games; gambling
Type
Research Article
Information
Advances in Applied Probability , Volume 49 , Issue 2 , June 2017 , pp. 388 - 410
Copyright
Copyright © Applied Probability Trust 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Albrecher, H. and Boxma, O. J. (2004). A ruin model with dependence between claim sizes and claim intervals. Insurance. Math. Econom. 35, 245–254. Google Scholar
[2] Asmussen, S. and Albrecher, H. (2010). Ruin Probabilities (Adv. Ser. Statist. Sci. Appl. Prob. 14), 2nd edn. World Scientific, Hackensack, NJ. Google Scholar
[3] Asmussen, S., Schmidli, H. and Schmidt, V. (1999). Tail probabilities for non-standard risk and queueing processes with subexponential jumps. Adv. Appl. Prob. 31, 422–447. Google Scholar
[4] Billingsley, P. (1995). Probability and Measure, 3rd edn. John Wiley, New York. Google Scholar
[5] Billingsley, P. (1999). Convergence of Probability Measures. John Wiley, New York. Google Scholar
[6] Bishop, D. T. and Cannings, C. (1978). A generalized war of attrition. J. Theoret. Biol. 70, 85–124. CrossRefGoogle ScholarPubMed
[7] De Moivre, A. (1710). De mensura sortis seu; de probabilitate eventuum in ludis a casu fortuito pendentibus. Phil. Trans. 27, 213–264. Google Scholar
[8] Dubins, L. E. and Savage, L. J. (1965). How to Gamble if You Must. Inequalities for Stochastic Processes. McGraw-Hill, New York. Google Scholar
[9] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. John Wiley, New York. Google Scholar
[10] Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Vol. I, 3rd edn. John Wiley, New York. Google Scholar
[11] Haigh, J. (1989). How large is the support of an ESS? J. Appl. Prob. 26, 164–170. CrossRefGoogle Scholar
[12] Hald, A. (2003). A History of Probability and Statistics and Their Applications before 1750. John Wiley, New York. Google Scholar
[13] Hines, W. G. S. (1981). Multispecies population models and evolutionarily stable strategies. J. Appl. Prob. 18, 507–513. Google Scholar
[14] Jacod, J. and Shiryaev, A. N. (1987). Limit Theorems for Stochastic Processes. Springer, Berlin. Google Scholar
[15] Kaigh, W. D. (1979). An attrition problem of gambler's ruin. Math. Mag. 52, 22–25. CrossRefGoogle Scholar
[16] Katriel, G. (2014). Gambler's ruin: the duration of play. Stoch. Models 30, 251–271. Google Scholar
[17] Kipnis, C. and Landim, C. (1999). Scaling Limits of Interacting Particle Systems. Springer, Berlin. Google Scholar
[18] Kozek, A. S. (1995). A rule of thumb (not only) for gamblers. Stoch. Process. Appl. 55, 169–181. Google Scholar
[19] Kurkova, I. and Raschel, K. (2015). New steps in walks with small steps in the quarter plane: series expressions for the generating functions. Ann. Combinatorics 19, 461–511. Google Scholar
[20] Mikosch, T. and Samorodnitsky, G. (2000). Ruin probability with claims modeled by a stationary ergodic stable process. Ann. Prob. 28, 1814–1851. CrossRefGoogle Scholar
[21] Raschel, K. (2014). Random walks in the quarter plane, discrete harmonic functions and conformal mappings. Stoch. Process. Appl. 124, 3147–3178. Google Scholar
[22] Ross, S. M. (1983). Stochastic Processes. John Wiley, New York. Google Scholar
[23] Smith, J. M. (1974). The theory of games and the evolution of animal conflicts. J. Theoret. Biol. 47, 209–221. CrossRefGoogle ScholarPubMed
[24] Smith, J. M. and Price, G. R. (1973). The logic of animal conflict. Nature 246, 15–18. CrossRefGoogle Scholar
[25] Stanley, R. P. (2011). Enumerative Combinatorics (Camb. Stud. Adv. Math. 49), Vol. I, 2nd edn. Cambridge University Press. CrossRefGoogle Scholar
[26] Taylor, P. D. (1979). Evolutionary stable strategies with two types of player. J. Appl. Prob. 16, 76–83. Google Scholar
[27] Thangavelu, S. (1993). Lectures on Hermite and Laguerre Expansions (Math. Notes 42). Princeton University Press. Google Scholar
[28] Uspensky, J. V. (1937). Introduction to Mathematical Probability. McGraw-Hill, New York. Google Scholar
[29] Van Leeuwaarden, J. S. H. and Raschel, K. (2013). Random walks reaching against all odds the other side of the quarter plane. J. Appl. Prob. 50, 85–102. Google Scholar
[30] Whitworth, W. A. (1901). Choice and Chance, with One Thousand Exercises. Hafner. Google Scholar
[31] Widder, D. (1946). The Laplace Transform. Princeton University Press. Google Scholar