Dynamic decisions in a laboratory setting.

• Author: Noussair, Charles
• Position: Optimal decision making

1. Introduction

   Many economic models involve a trade-off between current and future rewards. For example, in neoclassical one-sector growth models there is a trade-off between immediate consumption and investment for future consumption. In sequential job search models of labor economics, workers must decide between continuing their job search at a cost or accepting the best wage offered so far. In oligopoly models of tacit collusion, firms decide whether or not to defect from collusive behavior, and the defection leads to short-term benefit but future punishment. These models are all dynamic in the sense that an action at one stage influences the available actions or rewards at a future stage.

   Many such examples fall into the class of dynamic programming problems and for large classes of such problems, techniques for finding optima are well understood. Less well understood, however, are the decisions which individuals actually make in such dynamic settings. This is an empirical issue which lends itself to the methodology of laboratory investigation. There have already been a number of experiments in which subjects face a sequence of related decisions. Examples include the asset market studies of Camerer and Weigelt [3] and Smith, Suchanek, and Williams [13]. The predictions of dynamic game theory have been tested extensively, for example, in the contexts of the centipede game by McKelvey and Palfrey [9], of the repeated prisoner's dilemma by Selten and Stoecker [11] and Andreoni and Miller [1] and of bargaining by Ochs and Roth [10]. Dynamic job search has been studied by Cox and Oaxaca [4] and two armed bandits have been studied by Banks, Olson, and Porter [2].

   In the market and game theory studies the decisions made by subjects are complicated by the strategic aspects of the games; subjects' decisions must take into account the strategies they expect the other players to use. In asset markets, subjects seem to have difficulty making decisions and speculative bubbles frequently result. As noted by McCabe, Rassenti, and Smith [8], "The robust tendency of laboratory stock markets to produce bubbles is attributable to the myopic trading behavior of subjects. In effect, subjects fail to act according to the backward induction principle." The game-theoretic equilibrium refinement of subgame perfection, which depends on backward induction, fails to describe the observed outcomes in the game experiments cited above. The experiments in job search, in which the wage offers are stochastic, find that early termination of the search is observed (compared to a rational risk-neutral agent) though the data is consistent with the ability to solve complex optimization problems, if the presence of risk-aversion is postulated. In the two-armed bandit study, in which exogenous uncertainty is present, consistent deviations from the "maximizing" strategy, described by the authors as experimentation and hedging, are observed.

   In this paper we report data from an experimental study in which subjects are given a monetary incentive to solve a simple, non-stochastic, non-strategic, multi-stage, dynamic decision problem. The structure of the problem is motivated by the consumption vs. investment tradeoff of the one-sector growth problem. We structure the problem in very simple terms; subjects are asked to make ten decisions at discrete time intervals. There is no strategic uncertainty, no risk, no exogenous uncertainty, and no complex computation. These complicating features are removed from the decision problem. Decisions do not have to take into account beliefs about other players, there is no incentive for risk averse subjects to smooth out their payoffs, and no probability calculations need to be made. The optimum can be found using a simple rule. Under investigation is the empirical validity of a basic assumption of a variety of models such as those mentioned in the first paragraph. If subjects can successfully find the optimum for our decision problem, we need to consider the complicating elements as the source of the difficulty subjects have in the experiments cited above. If they cannot find the optimum, then one source of the difficulty may be purely in the sequential structure of the decision problems.

   The remainder of the paper is divided into sections as follows. In section II we discuss the structure of the decision problem and describe the experimental design. In section III we analyze the data. In section IV we give our concluding thoughts.

2. The Experiment

   In this section we describe the dynamic decision problem presented to the subjects.(1) For purposes of exposition we will describe the problem in the traditional terms of consumption and capital stock.(2) Subjects were faced with a discrete integer-valued approximation to the following dynamic decision problem:

   [Mathematical Expression Omitted] (1)

   subject to:

   [C.sub.t + 1] + [K.sub.t + 1] = f([K.sub.t]) + 0.5[K.sub.t] (2)

   [K.sub.t + 1], [C.sub.t + 1] [greater than or equal to] 0 (3)

   and

   [Mathematical Expression Omitted] (4)

   with t = 1, ..., 10 and [K.sub.1] given.

   [U.sub.t] is utility in each "round" t derived from consumption [C.sub.t], [K.sub.t] is the capital stock at time t, Q([K.sub.11]) is the utility from the remaining capital stock after time 10, f([K.sub.t]) is the production function and the existing capital stock depreciates by 50% each round. Consuming too much at any time drives the future capital stock down too much, and hence decreases future production and consumption. Investing too much at any time decreases utility from current consumption by too much. The optimal decision consists of a constant level of investment each round (except for the last round in some treatments). The optimal level of investment is independent of the utility function but does depend on the initial endowment.

   The experiments, which were computerized, took place at the CREED laboratory of the University of Amsterdam. The 48 participants were undergraduates at the university. Participants were not allowed to communicate with each other during the experiment. Each participant played 4 periods, numbered 0-3, each consisting of 10 rounds, for a total of 40 decisions. Each subject faced the same decision problem for all four periods. A participant's possibilities for investment in a given round depended on decisions he/she made in earlier rounds of the same period. The data from period 0 were discarded as they were for practice only and did not affect participants' final earnings. Participants took between 60 and 90 minutes to complete the experiment. Final earnings of subjects varied from 20 to 40 Dfl. (12-20 $U.S.). 8 Dfl. is a good hourly net wage for a college student in Amsterdam.

   Participants were given a table of production possibilities, Table XI in the appendix, that was a discrete approximation off ([K.sub.t]) + 0.5[K.sub.t]. The table lists, for any integer level of the capital stock in round t, all feasible combinations of consumption in round t and capital stock in round t + 1. Each participant was also given an initial endowment of capital stock (either 4 or 7) and a table (see tables XII and XIV in the appendix) of total utility associated with consumption in round t and corresponding marginal utilities, M[U.sub.t], in terms of "tokens," an experimental currency. The production function and the utility from consumption remained the same each round. Each subject's marginal utility was a discrete approximation of either

   [Mathematical Expression Omitted], (5)

   or

   [Mathematical Expression Omitted]. (6)

   Participants received "tokens" based on their consumption level. For example, if in a particular round, a subject with marginal utility [Mathematical Expression Omitted] consumed two units of C, the subject received 400/4 = 100 tokens for the first unit and 400/5 = 80 tokens for the second unit, or a total of 180 tokens for the round. These tokens were converted to Dutch guilders at the...