-optimal strategies. Mathematics of Operations Research 2 125-134.Since the second edition of Inequalities for Stochastic Processes: How to Gamble If You Must appeared in 1976, there have been many articles and several books inspired by the work of Dubins and Savage. This research includes work on specific gambling problems and questions raised in the book, and also contributions to the general theory of gambling and finitely additive probability theory.
The following list is necessarily incomplete in part because our knowledge of the literature is incomplete and in part because one must have a stopping rule.
Armstrong, Thomas E. (1980). Full houses and cones of excessive functions. Indiana University Mathematics Journal 29 737-746.
Armstrong, Thomas E. and Karel Prikry (1983). Singularity and absolute continuity with respect to strategic measures. Illinois Journal of Mathematics 27 624-658.
Armstrong, Thomas E. and William D. Sudderth (1981). Nearly strategic measures. Pacific Journal of Mathematics 94 251-257.
Banach, Stefan (1932). Théorie des Opérations Linéaire. Monografje Matematyczne, Vol. I, Warsaw.
Beam, John Eric (2002). Expectations for Coherent Probabilities. University of Miami (Thesis).
Berti, Patrizia and Pietro Riga (2004). Convergence in distribution of non-measurable random elements. Annals of Probability 32 365-379.
Bhaskara Rao, K. P. S. and M. Bhaskara Rao (1983). Theory of Charges: A Study of Finitely Additive Measures. Academic Press, New York.
Blackwell, David (1976). The stochastic processes of Borel gambling and dynamic programming. Annals of Statistics 4 370-374.
Blackwell, D. (1997). Large deviations for martingales. Festschrift for Lucien Le Cam, edited by David Pollard, Erik Torgersen and Grace L. Yang. Springer, New York 89-91.
Blackwell, D. and S. Ramakrishnan (1988). Stationary plans need not be uniformly adequate for leavable, Borel gambling problems. Proceedings of the American Mathematical Society 102 1024-1027.
Bodewig, Hans-Heinrich (1985). Markow-Entscheidungsmodelle mit Nichkon-vergenten Gesamtkosten. Universität Bonn (dissertation).
Chatterjee, Debasish, Eugenio Cinquemani and John Legeros (2011). Maximizing the probability of attaining a target prior to extinction. Nonlinear Analysis: Hybrid Systems 5 367-381.
Chen, May-Ru, Pei-Shou Chung and Yi-Cheng Yao (2008). On non-optimality of bold play for subfair red-and-black with a rational-valued house limit. Journal of Applied Probability 45 1024-1038.
Chen, May-Ru and Shoou-Ren Hsiau (2006). Two-person red-and-black games with bet-dependent win probability functions. Journal of Applied Probability 43 905-915.
Chen, Robert (1976/7). On almost sure convergence in a finitely additive setting. Zeitschrift fur Wahrscheinlichkeitstheorie und Verwandte Gebiete 37 341-356.
Chen, Robert and S. Ramakrishnan (1983). On the Glivenko-Cantelli theorem in a finitely additive setting. Sankhyā. The Indian Journal of Statistics. Series A 45 161-167.
Chen, Robert, Ilie Grigorescu and Larry Shepp (2011). Maximizing the discounted survival probability in Vardi’s casino. Stochastics 83 623-638.
Chen, Robert and Alan Zame (1979). On discounted primitive casino. Zeitschrift fur Wahrscheinlichkeitstheorie und Verwandte Gebiete 49 257-266.
Chen, Robert W., Larry A. Shepp, Yi-Ching Yao and Cun-Hui Zhang (2005). On optimality of bold play for primitive casinos in the presence of inflation. Journal of Applied Probability 42 121-137.
Chen, Robert W., Larry A. Shepp and Alan Zame (2004). A bold strategy is not always optimal in the presence of inflation. Journal of Applied Probability 41 587-592.
Dellacherie, Claude and Paul-André Meyer (1983). Théorie discrète du potentiel. Probabilités et potentiel. Chapitres IX à XI. Hermann, Paris.
Dellacherie, C. (1983-4). Quelques résultats sur les maisons de jeux analytiques. S´eminaire de Probabilit´es XIX, Lecture Notes in Mathematics 1123, Springer, New York 222-229.
Demko, Stephen and Theodore P. Hill (1981). Decision processes with totalcost criteria. Annals of Probability 9 293-301.
Demko, S. and T. P. Hill (1984). On maximizing the average time at a goal. Stochastic Processes and Their Applications 17 349-357.
Dubins, Lester E. (1982). A zero-one Borel Probability which admits of no countably additive extension. Proceedings of the American Mathematical Society 86 273-274.
Dubins, Lester E. (1998). Discrete red-and-black with fortune-dependent win probabilities. Probability in the Engineering and Informational Sciences 12 417-424.
Dubins, Lester E. (1999). Paths of finitely additive Brownian motion need not be bizarre. Séminaire de Probabilités XXXIII, Lecture Notes in Mathematics 1709, Springer, New York 395-396.
Dubins, Lester E. and William D. Sudderth (1977a). On countably additive gambling and optimal stopping theory. Zeitschrift fur Wahrscheinlichkeits-theorie und Verwandte Gebiete 41 59-72.
Dubins, Lester E. and William D. Sudderth (1977b). Persistently -optimal strategies. Mathematics of Operations Research 2 125-134.
Dubins, Lester E. and William D. Sudderth (1979). On stationary strategies for absolutely continuous houses. Annals of Probability 7 461-476.
Dubins, L., A. Maitra, R. Purves and W. Sudderth (1989). Measurable nonleavable gambling problems. Israel Journal of Mathematics 67 257-271.
Dubins, Lester E., Ashok P. Maitra and William D. Sudderth (2002). Invariant gambling problems and Markov decision processes. Handbook of Markov Decision Processes: Methods and Applications, edited by Eugene A. Feinberg and Adam Shwartz, Kluwer’s International Series 40 409-428.
Ethier, Stewart N. (2010). The Doctrine of Chances: Probabilistic Aspects of Gambling. Springer, New York.
Fainberg, E. A. (1982). Controlled Markov processes with arbitrary numerical criteria. Theory of Probability and its Applications XXVII 486-504.
Feinberg, Eugene A. (1996). On measurability and representation of strategic measures in Markov decision processes. Statistics, Probability and Game Theory: Papers in Honor of David Blackwell, edited by T.S. Ferguson, L.S. Shapley, and J.B. MacQueen, Institute of Mathematical Statistics Lecture Notes-Monograph Series 30 29-43.
Fernholz, Daniel and Ioannis Karatzas (2011). Optimal arbitrage under model uncertainty. Annals of Applied Probability 21 2191-2225.
Gangopadhyay, Sreela and B. V. Rao (1997). Some finitely additive probability: random walks. Journal of Theoretical Probability 10 643-657.
Gilat, David and William Sudderth (1977). Timid play when large bets are profitable. Annals of Probability 5 573-576.
Gilat, David and Ernst-August Weiss. Jr. (1976). On roulette which allows stakes on infinitely many holes. Israel Journal of Mathematics 24 282-285.
Grigorescu, Ilie, Robert Chen and Larry Shepp (2007). Optimal strategy for the Vardi casino with side payments. Journal of Applied Probability 44 199-211.
Hill, Theodore Preston (1979). On the existence of good Markov strategies. Transactions of the American Mathematical Society 247 157-176.
Hill, Theodore P. and Victor C. Pestien (1983). The advantage of using nonmeasurable stop rules. Annals of Probability 11 442-450.
Hill, Theodore P. and Victor C. Pestien (1987). The existence of good Markov strategies for decision processes with general payoffs. Stochastic Processes and Their Applications 24 61-76.
Hordijk, A. (1977). Dynamic Programming and Markov Potential Theory. Mathematical Centre Tracts 51, Mathematisch Centrum, Amsterdam.
Karandikar, R. L. (1982). A general principle for limit theorems in finitely additive probability. Transactions of the American Mathematical Society 273 541-550.
Karandikar, Rajeeva L (1988). A general principle for limit theorems in finitely additive probability: the dependent case. Journal of Multivariate Analysis 24 189-206.
Karatzas, Ioannis and Daniel Ocone (2002). A leavable bounded-velocity stochastic control problem. Sochastic Processes and Their Applications 99 31-51.
Karatzas, Ioannis and Ingrid-Mona Zamfirescu (2008). Martingale approach to stochastic differential games of control and stopping. Annals of Probability 36 1495-1527.
Klugman, Stuart (1977). Disccounted and rapid subfair red-and-black. Annals of Statistics 5 734-745.
Kulldorff, Martin (1993). Optimal control of favorable games with a time limit. SIAM Journal of Control and Optimization 31 52-69.
Laraki, Rida and William D. Sudderth (2004). The preservation of continuity and Lipschitz continuity by optimal reward operators. Mathematics of Operations Research 29 672-685.
Lou, Jianxiong (2009). Gambling Theory and Stock Option Models. Rutgers University (Thesis).
Łoś, J. and E. Marczewski (1949). Extensions of measure. Fundamenta Mathematicae 36 267-276.
Luxemburg, W. A. J. (1991). Integration with respect to finitely additive measures. Positive Operators, Riesz Spaces, and Economics: Proceedings of a Conference at Caltech, April 16-20, 1990, edited by Charalambos D. Aliprantis, Kim C. Border, and Wilhelmus A. J. Luxemburg. Springer-Verlag, Berlin 109-150.
Maitra, A., R. Purves and W. Sudderth (1990). Leavable gambling problems with unbounded utilities. Transactions of the American Mathematical Society 320 543-567.
Maitra, A., R. Purves and W. Sudderth (1991a). A Borel measurable version of König’s lemma for random paths. Annals of Probability 19 423-451.
Maitra, A., R. Purves and W. Sudderth (1991b). A capacitability theorem in finitely additive gambling. Rendicotti di Matematica e delle sue applicazioni Ser VIII 11 819-842.
Maitra, A., R. Purves and W. Sudderth (1992a). A capacitability theorem in countably additive gambling. Transactions of the American Mathematical Society 333 221-249.
Maitra, A., R. Purves and W. Sudderth (1992b). Approximation theorems for gambling problems and stochastic games. Game Theory and Economic Applications. Lecture Notes in Economics and Mathematical Systems 389. Springer-Verlag, Berlin 114-132.
Maitra, Ashok, Victor Pestien and S. Ramakrishnan (1990). Domination by Borel stopping times and some separation properties. Fundamenta Mathematicae 135 189-201.
Maitra, Ashok P. and William D. Sudderth (1989). An operator solution of stochastic games. Israel Journal of Mathematics 78 33-49.
Maitra, Ashok P. and William D. Sudderth (1993). Borel stochastic games with lim sup payoff. Annals of Probability 21 861-885.
Maitra, Ashok P. and William D. Sudderth (1996a). Discrete Gambling and Stochastic Games. Springer-Verlag, New York.
Maitra, Ashok P. and William D. Sudderth (1996b). The gambler and the stopper. Statistics, Probability and Game Theory: Papers in Honor of David Blackwell, edited by T.S. Ferguson, L.S. Shapley, and J.B. MacQueen, Institute of Mathematical Statistics Lecture Notes-Monograph Series 30 191-208.
Maitra, Ashok P. and William D. Sudderth (1998). Finitely additive stochastic games with Borel measurable payoffs. International Journal of Game Theory 27 257-267.
Maitra, A. and W. Sudderth (1999). An introduction to gambling theory and its applications to stochastic games. Annals of the International Society of Dynamic Games 4 251-269.
Maitra, A. and W. Sudderth (2000a). Saturations of gambling houses. Séminaire de Probabilités XXXIV. Lecture Notes in Mathematics 1729. Springer, New York 218-238.
Maitra, A. and W. Sudderth (2000b). Randomized strategies and terminal distributions. Game theory, Optimal Stopping, Probability and Statistics: Papers in Honor of Thomas S. Ferguson, edited by F. Thomas Bruss and Lucien Le Cam, Institute of Mathematical Statistics Lecture Notes-Monograph Series 35 39-52.
Meilijson, Isaac (to appear). The re-opening of Dubins & Savage casino in the era of diversification. Probability in the Engineering and Informational Sciences.
Monticino, Michael G. (1991a). The adequacy of universal strategies in analytic gambling problems. Mathematics of Operations Research 16 21-41.
Monticino, Michael G. (1991b). Utility functions which ensure the adequacy of stationary strategies. Transactions of the American Mathematical Society 325 187-204.
Mountford, T. S. (1993). A solution to a problem of Dubins and Savage. Israel Journal of Mathematics 83 253-256.
Pendergrass, Marcus and Kyle Siegrist (2001). Generalizations of bold play in red and black. Stochastic Processes and Their Applications 92 163-180.
Pestien, Victor C. (1982). An extended Fatou equation and continuous-time gambling. Advances in Applied Probability 14 309-323.
Pestien, Victor C. (1983). Weak approximation of strategies in measurable gambling. Pacific Journal of Mathematics 109 157-164.
Pestien, Victor C. and William D. Sudderth (1985). Continuous-time red and black: how to control a diffusion to a goal. Mathematics of Operations Research 10 599-611.
Pestien, Victor C. and William D. Sudderth (1988). Continuous-time casino problems. Mathematics of Operations Research 13 364-376.
Pestien, Victor and Xiaobo Wang (1993). Finite-stage reward functions having the Markov adequacy property. Stochastic Processes and Their Applications 46 129-151.
Pontiggia, Laura (2005). Two-person red-and-black with betdependent win probabilities. Advances in Applied Probability 37 75-89.
Pontiggia, Laura (2007). Non-constant sum red-and-black games with bet-dependent win probability function. Advances in Applied Probability 44 547-543.
Purves, R. A. and W. D. Sudderth (1976). Some finitely additive probability. Annals of Probability 4 259-276.
Purves, Roger A. and William D. Sudderth (1982). How to stay in a set or König’s lemma for random paths. Israel Journal of Mathematics 43 139-153.
Purves, Roger A. and William D. Sudderth (1983). Finitely additive zero-one laws. Sankhyā. The Indian Journal of Statistics. Series A 45 32-37.
Purves, Roger A. and William D. Sudderth (2010). Big Vee: the story of a function, an algorithm, and three mathematical worlds. Sankhyā. The Indian Journal of Statistics. Series A 72 37-63.
Ramakrishnan, S. (1981). Finitely additive Markov chains. Transactions of the American Mathematical Society 265 247-272.
Ramakrishnan, S. (1983). The tail σ-field of a finitely additive Markov chain starting from a recurrent state. Proceedings of the American Mathematical Society 89 493-497.
Ramakrishnan, S. (1984a). Potential theory for finitely additive Markov chains. Stochastic Processes and Their Applications 16 287-303.
Ramakrishnan, S. (1984b). Central limit theorems in a finitely additive setting. Illinois Journal of Mathematics 28 139-161.
Ramakrishnan, S. (1986). A finitely additive generalization of Birkhoff’s ergodic theorem. Proceedings of the American Mathematical Society 96 299-305.
Ramakrishnan, S. and W. Sudderth (1988). A sequence of coin toss variables for which the strong law fails. American Mathematical Monthly 95 939-941.
Ramakrishnan, S. and W. Sudderth (1998). Geometric convergence of algorithms in gambling theory. Mathematics of Operations Research 23 568-575.
Rieder, U. (1976). On optimal policies and martingales in dynamic programming. Journal of Applied Probability 13 507-518.
Ruth, Kevin (1999). Favorable red and black on the integers with a minimum bet. Journal of Applied Probability 36 837-851.
Ross, Sheldon M. (1974). Dynamic programming and gambling models. Advances in Applied Probability 6 598-606.
Schäl, Manfred (1975). Conditions for optimality in dynamic programming and for the limit of n-stage optimal policies to be optimal. Zeitschrift fur Wahrscheinlichkeitstheorie und Verwandte Gebiete 32 179-196.
Schäl, Manfred (1989). On stochastic dynamic programming: a bridge between Markov decision processes and gambling. Markov Processes and Control Theory, edited by H. langer and V. Nolkau. Mathematica Research 54. Akademie-Verlag, Berlin 178-216.
Schäl, Manfred (1990). On the chance to visit a goal infinitely often. Optimization 21 585-592.
Schweinsberg, Jason (2005). Improving on bold play when the gambler is restricted. Journal of Applied Probability 42 321-333.
Secchi, Piercesare (1997). Two-person red-and-black stochastic games. Journal of Applied Probability 34 107-126.
Shepp, L. (2006). Bold play and the optimal policy for Vardi’s casino. Random Walk, Sequential Analysis and Related Topics: A Festschrift in Honor of Yuan-Shih Chow, edited by Agnes Chao Hsiung, Zhiliang Ying and Cun-Hui Zhang, World Scientific Publishing Co. 150-156.
Sudderth, William D. (1983). Gambling problems with a limit inferior payoff. Mathematics of Operations Research 8 287-297.
Sudderth, William D. and Ananda Weerasinghe (1989). Controlling a process to a goal in finite time. Mathematics of Operations Research 14 400-409.
Wagner, Daniel H. (1977). Survey of measurable selection theorems. SIAM Journal on Control and Optimization 15 859-903
Wagon, Stan (1985). The Banach-Tarski Paradox. Cambridge University Press, Cambridge, United Kingdom
Yao, Yi-Cheng (2007). On optimality of bold play for discounted Dubins-Savage gambling problems with limited playing times. Journal of Applied Probability 44 212-225
Yao, Yi-Cheng and May-Ru Chen (2008). Strong optimality of bold play for discounted Dubins-Savage gambling problems with time-dependent parameters. Journal of Applied Probability 45 403-416