Risk and Return Considerations in “The Weakest Link”
B. Ross Barmish, Nigel Boston · American Mathematical Monthly · 2009
The television game show, The Weakest Link, involves contestants making a sequence of decisions over time. Given the rules of the game and the process for accrual of payoffs over time, a number of authors have recognized that this show serves as a laboratory for assessment of human decision-making. To this end, by comparing theoretically derived gaming strategies with those actually used by the contestants, conclusions are drawn regarding the extent to which players ’ decisions are rational and consistent with the pursuit of optimality. The first main objective of this paper is to provide arguments that the models used in the literature to date may result in an erroneous impression of the extent to which contestants ’ decisions deviate from the optimum. More specifically, we first point out that previous authors, while concentrating on maximization of the expected value of the return, totally neglect the risk component; i.e., the expected return is considered while its variance is not. To this end, we expand the analysis of previous authors to include both risk and return and a number of other factors: mixing of strategies and socalled end effects due to fixed round length. It is seen that many strategies, discounted by previous authors as being sub-optimal in terms of maximization of expected return, may in fact be consistent with rational decision-making. That is, such strategies satisfy a certain “efficiency ” requirement in the risk-return plane. To obtain this efficiency characterization of the game, the paper also includes results describing mean and variance of the return in closed form. 1.