Randomized Strategies and Prospect Theory in a Dynamic Context

Transcription

1 Randomized Strategies and Prospect Theory in a Dynamic Context Vicky Henderson University of Warwick & Oxford-Man Institute David Hobson University of Warwick Alex S.L. Tse University of Warwick February 6, 5 Applying prospect theory (PT) in a dynamic context brings new challenges. We study PT agents who face optimal timing decisions in a dynamic setting. It is known that the probability weighting aspect of PT leads to time inconsistency. It also leads to another feature which has not been considered to date - agents may prefer randomized strategies to pure strategies. In this paper we consider the impact of allowing PT agents facing dynamic timing decisions to follow randomized strategies. In the discrete model of casino gambling of Barberis () we show that allowing randomized strategies leads to significant gains in PT value. In the continuous model of Ebert and Strack (4) we show that allowing randomization can significantly alter the predictions of the model. Ebert and Strack show that a naive investor with PT preferences but without the ability to follow randomized strategies, never chooses to stop and gambles until the bitter end. We show that this extreme conclusion is no longer valid if the agent has a coin in his pocket. Keywords: Behavioral economics, prospect theory, probability weighting, randomized strategies JEL Classification: D3, D8, G. First version: November 7, 4. The authors would like to thank Philipp Strack, Peter Wakker, delegates at QMF, Sydney (Dec 4), and seminar participants at the Judge Business School, University of Cambridge (Jan 5) and the DR@W Seminar, University of Warwick (Feb 5) for helpful comments. University of Warwick, Coventry, CV4 7AL. UK and Oxford-Man Institute, University of Oxford, Eagle House, Walton Well Rd, Oxford. OX 6ED, UK. vicky.henderson@warwick.ac.uk University of Warwick, Coventry, CV4 7AL. UK. d.hobson@warwick.ac.uk University of Warwick, Coventry, CV4 7AL. UK. S.L.Tse@warwick.ac.uk

2 Despite the attractiveness of the principles of expected utility theory (EUT), it has long been recognized that it fails to fully explain individuals attitudes towards risk. One of the most prominent alternatives to EUT is prospect theory (PT) (Tversky and Kahneman (99)), an extention of the original prospect theory proposed by Kahneman and Tversky in 979. PT postulates that gains and losses are evaluated relative to a reference point using a function which is concave over gains but convex on losses, and steeper for losses than for gains to capture loss aversion. The most distinctive feature of PT is that cumulative probabilities are transformed such that individuals overweight tail events (as proposed by Quiggin (98)). Whilst probability weighting has been well studied in static one-period models, the challenges involved in extending the theory to a dynamic setting are considerable, and our understanding of the implications in a dynamic context is still developing. The main challenge is that in a dynamic setting, probability weighting may - and typically will - induce time inconsistency. Time inconsistency predicts that how an individual acts in a state of the world may differ from how he planned to act in that state. It becomes crucial to understand how to deal with the time inconsistency - see Machina (989) - is the agent naive (and thus unaware of the inconsistency), or is he sophisticated and aware of the inconsistency? And if he is sophisticated, can he find a way to commit to his initial plan, or must he acknowledge the fact that his future self will re-optimize, and potentially change strategy? In his recent discussion paper on the psychology of tail events, Barberis (3a) raises the challenge of how to best address the time inconsistency induced by probability weighting. He observed and investigated this phenomenon in his model of casino gambling in a multi-period discrete time setting, see Barberis (). The standard paradigm is to restrict agents to pure strategies 3. In this paper we consider the role of randomization in the agent s set of potential strategies and re-examine the impact of In contrast there is a huge literature concerned with the time inconsistency of non-exponential discounting, in particular hyperbolic discounting, and a growing literature on more general dynamic consumption and asset allocation models (see for example, Harris and Laibson (), Basak and Chabakauri () and the discussion of the classic paper of Strotz (956).) In a continuous time setting, Xu and Zhou (3) study a sophisticated PT agent with a committment device that allows him to stick to his initial plan. 3 There are only limited results on mixed strategies. One exception is Theorem 7.4. of Wakker (), which considers mixed strategies in the context of a static choice of prospect in a rank dependent utility setting, and relates the benefit of using mixed strategies to the concavity of the probability weighting function.

3 probability weighting in a dynamic context. 4 The first contribution of our paper is to highlight that a PT agent facing a dynamic investment/stopping problem will typically benefit from following a randomized strategy and will alter his optimal behavior both at time zero and at later times 5. We demonstrate this result in the context of the casino gambling model of Barberis (). In this setting the wider choice of available strategies causes the agent to gamble in a larger regime of parameters. The second contribution is to re-consider the model of Ebert and Strack (4), which is the continuous analog of Barberis s () discrete model. Ebert and Strack reach a very striking conclusion that a naive agent with PT preferences never stops in this model instead he gambles until the bitter end. We show that this strong conclusion is not valid for a naive agent with access to randomization. Such an agent has many optimal strategies, some of which involve stopping at the reference level. In the sense that the results of Ebert and Strack cast doubt on the applicability of PT in a dynamic context, our work resurrects the approach there are combinations of value function and probability weighting functions for which the PT agent follows a non-trivial strategy. In the context of the casino gambling model of Barberis (), the ability of agents to design stopping rules which involve continuing in states of profit, but stopping in loss states, means they can synthesize lottery-like payoffs, even in a symmetric binomial tree, which is very attractive to the PT agent. In particular, the PT agent chooses to play in the casino (enter the game) in 4 In November 4, we became aware through a conversation with one of the authors, that He et al (4) had also been studying the impact of randomization on the strategies of PT agents in optimal stopping problems. First public versions of both papers were posted to SSRN later that year. Both this paper and He et al (4) begin with the observation that in a discrete setting PT agents who can choose a probability of continuing outperform those constrained to make a binary choice between stopping and continuing. Thereafter the two papers differ in emphasis. He et al (4) focus on the agent who can pre-commit in a discrete-time, infinite-horizon model. In contrast, our focus is on the implications for the continuous-time setting, and especially the question of whether naive agents always use a trivial strategy in this setting. 5 Although we consider an agent with PT preferences, similar arguments will apply in any setting where probability weighting is included. In particular, there will be benefits from following randomized strategies in dynamic stopping models under the rank-dependent utility (RDU) models of Quiggin (98) and Yaari (987). 3

4 situations where EUT would predict he would not enter. 6 One of the conclusions of Barberis () is that PT provides a tenable model of gambling in a casino, since it provides a theoretical basis within which agents do indeed choose to gamble. For an agent in the Barberis () model, a strategy is a rule which in each state, on each future date, determines whether to continue gambling or not. If the agent chooses not to gamble then they cannot reverse this decision later - their visit to the casino is deemed to be over. The outcomes of gambles are either a win or a loss, with a corresponding up or down move in a binomial tree. An agent evaluates the expected outcomes using PT preferences. Barberis introduces three types of agent: a sophisticated agent, an agent with a pre-commitment mechanism (for short, an agent with commitment) and a naive agent. The first agent is a sophisticated agent - an agent who not only knows and understands his full preference structure, but also knows that his future self understands this preference structure. Therefore this agent takes decisions to continue or to stop gambling in the complete knowledge of how his future self will behave in any of the scenarios in which he continues gambling. The problem of solving for the optimal behavior of the sophisticated agent is in many respects the simplest of the three types, since it can be determined by backward induction. The second agent is able to commit his future self to following a given strategy. The agent with commitment decides on his strategy today (using PT preferences, and based on current probabilities) and has a mechanism for making his future self follow this strategy. For example, if this agent wants to follow a loss-exit strategy, then he can impose this strategy on his future self by ensuring he has no opportunity to raise further funds later in the evening (eg. by taking cash to the casino and leaving cards at home). The final agent is naive. At the current time he evaluates prospects in an identical fashion to the agent with commitment. But if today he elects to gamble 7, with the intention of following a given strategy in the future, then he has no way of forcing his future self to follow that strategy. Instead, at each future moment in time the agent re-evaluates all possible strategies and takes a new decision about whether to gamble further, or to stop. 6 EUT predicts a risk averse agent would never enter a casino after all, bets are risky and have negative return unless some modifications are made to the theory, eg. social benefits of an evening with friends, or the excitement of waiting for the outcomes of bets to unfold. 7 If today he elects not to gamble, then this decision is irreversible, and he is deemed to have left the casino. 4

5 Barberis () solves for the optimal behavior of the three types of agent in a binomial tree over five time periods. He finds PT can explain observed patterns of gambling behavior. Each type of agent will, for certain values of PT preferences, gamble in a casino. The first contribution of Barberis () is to show that PT provides a viable model of gambling behavior. The second major contribution of Barberis () is to make a striking observation concerning the behavior of the naive agent. Suppose parameters are such that the naive agent wants to gamble at time zero. On first entering the casino, the naive agent typically plans to continue to gamble if his first bet is a success, but to stop if his first bet is unsuccessful. Such a strategy is a stop-loss or loss-exit strategy. But what happens in practice? If the first gamble results in a win, then the agent re-evaluates his situation and may well find it optimal to stop. In particular, the extreme outcome of a winning gamble in every time period up to the final horizon is now less extreme, and thus the impact of probability weighting is less signficant. Then the fact that the agent is risk averse (on gains) means that he stops gambling. Conversely, if the first gamble results in a loss, the agent re-evaluates, and again the probability weighting is less significant, resulting in the agent electing to continue gambling (as PT preferences are risk seeking on losses). Thus, Barberis () finds that sometimes the naive agent follows a strategy which is the exact opposite of the one he planned to follow at the outset. Now consider the continuous-time, continuous-space analog of the Barberis () model in which wealth (prior to any decision to stop gambling) follows a Brownian motion. Agents choose a stopping time, so that the payoff is the value of the stopped process; agents evaluate payoffs using PT. This is the model considered in Ebert and Strack (4). Ebert and Strack show that for a wide range of PT specifications (including essentially all versions which have received empirical support) the agent with commitment would prefer a stopping rule based on first exit time of Brownian motion from a well-chosen interval, where the current value lies strictly inside this interval, to one of stopping immediately. From this fact Ebert and Strack make a drastic inference: the naive agent never stops; instead he always finds an alternative strategy which is preferable to stopping. What is the contribution of our paper? Consider first the discrete-time setting. Our contribution is to argue that PT agents (of all three types) can benefit from following randomized strategies. Barberis () is careful to exclude these strategies 8, but unlike in the EUT 8 Ebert and Strack (4) do not explicitly exclude randomized strategies but they are implicitly excluded by 5

6 paradigm 9 an agent who randomizes his strategy can outperform an agent who does not. We give a complete analysis for the casino gambling model for two periods. We first show that agents who can randomize are more likely to gamble at time zero (in the sense that there are more parameter sets for which such agents choose to gamble, when compared with agents of the same type, who cannot randomize) and second that the value functions of agents who can randomize strictly dominate those of agents who cannot. Is this contribution significant? To argue that it is significant, first, we need to argue why it is plausible that agents should be allowed to randomize their strategies and second, we need to demonstrate that this has a sizeable impact on the behavior of agents. This is the plan for the first half of the paper. In the next section, we argue that it is plausible to allow for randomization. Then, instead of three classes of agents there are six - three types (sophisticated, with committment, naive) each occuring in two flavors (unable to randomize, able to randomize). In Section we describe the model in detail. Section 3 considers two periods and specializes to the Tversky and Kahneman (99) PT and investigates numerically the dependence of each agent s optimal strategy on the probability weighting, risk aversion/seeking and loss aversion parameters. Now consider the continuous time setting. In this case our contribution is to show that a naive agent with a randomization device (eg. a coin) and with the ability to commit his current self to following strategies which depend on the realization of this device (eg. continuing if the coin lands heads, and stopping otherwise) does not necessarily follow the extreme never-stopping strategy proposed by Ebert and Strack (4). Given initial wealth x (our focus is on the case x = r where r is the reference level), Ebert and Strack show (in a very general PT framework) how to choose an interval [a, b] with a < x < b, such that stopping at the first exit time from this interval is strictly preferred to stopping immediately. In Section 4 we show (with only a few extra assumptions, all of which are satisfied in standard versions of PT) that the randomized strategy of sometimes stopping immediately and sometimes stopping on first exit from [a, b] is a better prospect than simply stopping on first exit from the interval. Hence the analysis of Ebert and Strack is not sufficient to conclude that the naive agent never stops, if randomization is allowed. their assumption that the initial σ-algebra only contains events of probability zero or one. 9 In the EUT paradigm, the expected payoff from a mixed strategy is simply the appropriate mixture of expected payoffs of the component strategies. Hence in the EUT setting there is a pure strategy which is optimal. 6

7 Ebert and Strack (4) prove that for agents restricted to pure strategies, the optimal strategy is to gamble until the bitter end. As argued above, their proof does not apply to agents with access to randomized strategies. But perhaps their result is true anyway, in the sense that the best strategy (for naive agents who may randomize) involves never stopping immediately. In Section 5 we show this is not the case. For a stylized value function (and for a wide class of probability weighting functions) we calculate the optimal strategy of a naive agent who can randomize. We show that if the initial value of the Brownian motion is at the reference level, then the optimal prospect includes an atom at the reference level. There are many ways of achieving this prospect, but some of them involve stopping at the initial time. Hence, an naive agent who can randomize may stop at the reference level, and PT preferences in a dynamic setting can lead to non-trivial behavior. To illustrate that our analysis is not restricted to Brownian motion and this stylized value function, and to ensure that it is as wide ranging as that of Ebert and Strack () we also consider drifting Brownian motion and a shifted-power value function. We conclude that for value functions with finite (marginal) loss aversion, the optimal prospect includes weight at the reference level, and thus, a naive agent may stop there. If a naive agent can follow strategies which depend on exogenous events, then he may realize this target by stopping at the reference level and thus the dramatic conclusion of Ebert and Strack () that naive agents never stop does not hold. PT has already enjoyed considerable success in its application to finance. In a static setting, PT has been used to capture the preference for lottery-like payoffs which are positively skewed. More broadly, this idea has been used to shed light on many empirical puzzles in financial markets, and, in particular, the equity premium puzzle. The S shaped form of the value function in PT (in the absence of probability weighting) has been applied successfully in dynamic timing models in finance. Many of these studies have aimed at shedding light on the disposition See in particular, the recent review papers of Barberis (3a, 3b). Barberis and Huang (8) look at the implications of probability weighting for stock prices and show skewness is priced. See Benartzi and Thaler (995), Barberis, Huang and Santos () and De Giorgi and Legg (). Other financial market applications of PT include to explain the low average returns of IPO stocks, stocks traded in over-the-counter markets and high-volatility stocks, the apparent overpricing of out-of-the-money options, and the lack of diversification in many household portfolios. 7

8 effect - the widespread phenomena whereby gains are realized more often than losses. 3 However, the study of dynamic models with probability weighting, in particular, dynamic timing models, has been limited. Building upon the insightful contributions of Barberis () and Ebert and Strack (4), our paper is a further attempt to improve our understanding of prospect theory in a dynamic context. Can Agents follow Randomized Strategies? Is it plausible to allow agents in a casino to follow randomized strategies? Consider first the sophisticated agent. This agent is sufficiently self-aware to understand how his future self will operate; it seems illogical that he could not and would not use randomization if it were possible. How might randomization be achieved? It could depend on the outcome of an event at a neighbouring table, or which hand the croupier uses, or which host will appear with drinks, or whether the casino offers a voucher for a meal. Suppose for example the agent has a coin in his pocket, or the casino chip has distinguishable sides. The agent can decide to continue if and only if the coin lands heads, or if the result of seven spins is at least five heads. 4 Even if we do not allow external randomization, we should allow the possibility of an agent deciding whether to continue gambling based on his betting history. For example, we should allow an agent to continue to gamble at time two if his first two bets were a win followed by a loss, but not to continue if they were a loss followed by a win (or vice versa) 5. 3 For example, Kyle, Ou Yang and Xiong (6) and Henderson () study one-shot timing models, whilst more recently, Barberis and Xiong () and Ingersoll and Jin (3) have made advances in studying multiple stopping (reinvestment) problems. All of these papers prove the optimality first exit strategies, whereby investors sell or exit when the price reaches either of two threshold levels. 4 Anecdotally then, this provides an explanation for the practice of deciding to perform an action if the outcome of some completely independent and random event is a success. Psychologically, the rationale for introducing dependence on a random event is to result in a decision being taken - at which point it is easier to decide if the decision feels appropriate, and if not, the decision is revisited. But prospect theory provides an alternative interpretation - agents can do better with randomized strategies and hence should spin a coin to help reach an optimal solution. 5 For this example, we are presuming the agent planned to continue irrespective of the outcome of the first bet. In general, there will be some node which can be reached by different histories of wins and losses and then the decision to continue can be made in such a way that it depends on the wealth history. Viewed from time zero, this corresponds to a continuation probability which is neither zero nor one. 8

9 What about the agent with the ability to commit to his future self? It would seem plausible that such an agent could commit his future self to continue gambling if the outcome of some random event fell in some set, and otherwise to stop. By appropriate design it ought to be possible to choose the set of outcomes which lead to continuation to have the desired probability. What about the naive agent? Here, perhaps the conclusion is less clear. Does it make sense for a naive agent to randomize his strategy? In particular, if the agent is truly naive, then they are not bound by the outcomes of any past events, and they need not be bound by the outcome of whichever event is proposed to be used to determine their strategy. But alternatively, it can be argued that although agents are naive, they are not stupid. If they know they can benefit from randomizing their strategy then they should do so. After all, they are not committing their future selves to an action, but rather their current self. Finally then, it comes down to a modeling choice and for the naive agent and either choice is defensible. Ultimately the pragmatic reason that agents can perform better with randomized strategies is a justification for considering them. 6 What about in continuous time? In that case our focus is on the naive agent. Can such an agent randomize? In particular, can such an agent stop immediately with probability θ (, )? In the set-up of Ebert and Strack (4) the filtration is generated by the Brownian motion, in which case all events at the initial time have probability zero or one and randomization is not possible. But, equally, it is possible to work in a richer probability space supporting, at a minimum, a uniform random variable. As in the discrete time case it seems reasonable that the agent might have a coin in his pocket, and that he can make his strategy dependent on the outcomes of spins of this coin. The standard paradigm is to consider only pure decision rules, but we show that the conclusions are not robust to a change of setting to include randomized strategies. 6 Two reasons for not allowing randomization are first that it is not a usual assumption in the behavioral economics literature, and second, in allowing path-dependent and randomized strategies, we must expect to lose uniqueness of the optimal strategy. We thank Philipp Strack for this observation. 9

10 The Discrete Model. Wealth dynamics An agent is offered a series of independent and identical gambles where he could win or lose an amount of h with equal probability in each gamble. The time horizon consists of T periods. At the beginning of each period, the agent could decide whether to enter the gamble (continue), or to leave (stop) the whole game and take the current cumulative profit or loss as the final payoff. At the end of the T th period, no more potential gambles are available and the whole game ends. The evolution of the agent s net worth can be represented by a path through a T - period recombining binomial tree. At each time point t =,,..., T, his net worth can possibly take t + distinct values labeled by S(, t) > S(, t) >... > S(t +, t). If h =, and if the initial wealth of the agent is zero then S(k, t) = t (k ). Henceforth we assume h =, the results for other values can be recovered from scaling.. Prospect Theory preferences The final outcome of the entire game can be summarized by a random variable X which is the net profit or loss to the agent when the game terminates. The agent s preference is described by PT. Consider a prospect represented by a discrete random variable X taking values x m < x m+ < < x < x = < x < < x n < x n (so that x m, x m+,... x correspond to losses and x,... x n correspond to gains, note that we are taking the reference level to be zero) with probabilities p i = P(X = x i ), i = m, m +,..., n. We write this prospect as P = P X where P X = (x m, p m ; x m+, p m+ ;... ; x, p ; x, p ; x, p ;... ; x n, p n ). The prospect P X is valued as n V (X) = [w + (p n + p n +... p i ) w + (p n + p n +... p i+ )]v(x i ) () i= + [w (p m + p m p i ) w (p m + p m p i )]v(x i ), i= m where the value function v : (, ) (, ) is increasing, and w ± are a pair of probability weighting functions w ± : [, ] [, ] which are increasing and continuous and satisfy w ± () = and w ± () =.

11 Unless otherwise stated we assume that the value function is such that v is continuous, v() =, v is concave and increasing on x > and convex and increasing on x < and v satisfies the (simple) loss aversion property v(x) + v( x) < for all x. We will call the case where v( x) = kv(x) (for x > ), with k >, the scaled rotation case. 7 In Section 5, we will use the notation k = + c for c > for convenience. Unless otherwise stated we will assume that w ± is differentiable on [, ], and concave on [, q ± ] and convex on [q ±, ] for some q. If w + = w we will write w for both functions. Sometimes we will require additional properties, such as w( ) ; where required, these additional properties will be explicitly stated.8.3 Deterministic Optimal Strategies Barberis () assumes that the agent s decision at a particular node is a binary choice of exit or continuation which only depends on the current position of that node. A plan at node (i, t) is defined as a mapping C : (i, t) {stop, continue}. A stopping strategy is a collection of plans, one for each node which is reachable from the starting node. Under a fixed stopping strategy one can compute the probability distribution of X a, where X a is the net profit or loss on the termination of game by following strategy a. The PT value of this strategy can be computed by ()..3. The Pre-committing Agent At time zero, the agent maximizes the PT value V (X a ), where the maximum is taken over stopping strategies based on the starting node (i =, t = ). An agent is said to be pre-committing 7 The Kahneman and Tversky (979) value function takes v(x) = x α + for x > and v(x) = k x α for x <. () If α + = α then this is a scaled rotation. 8 The Tversky and Kahneman (99) probability weighting function is the choice p δ ± w ±(p) = (p δ ± + ( p) δ ±) /δ ± (3) for parameters.3 < δ ± < and which does have the property w ±( ). The lower bound on δ is required to ensure that w ± is increasing. Often we will assume that w + = w, or equivalently δ + = δ. The Goldstein and Einhorn (987) probability weighting function is the choice w(p) = ζp δ for < ζ < and < δ <. We (ζp δ +( p) δ ) can specify different weighting functions for gains and losses by making ζ or δ depend on whether we are in the gain or loss regime. The choice ζ < ensures that w( ).

12 if he acts according to the stopping strategy implicit in the maximizer in any subsequent node (i, t). In this case, the optimal strategy is only computed at time zero and it characterizes the subsequent behaviors of the agent completely..3. The Naive Agent Unlike the pre-committing agent, the naive agent re-computes the optimal strategy at every node of (i, t) and decides whether to continue or stop in the current node based on the new optimization result. The updated decision overrides the plans derived in any previous time step. The agent is said to be naive to highlight the time-inconsistent behaviors due to the possible change in plans at some nodes..3.3 The Sophisticated Agent This agent is aware that his future self will re-evaluate the prospect theory value in future states, and in the future will follow the best strategy at that moment in time. This type of agent reasons by the logic of backward induction. Starting from the last period of the game, the agent solves for the optimal plan at nodes of (, T ), (, T ),..., (T, T ). Knowing how he will behave in the future time, he iterates one time-step backward and decides whether it is optimal to continue or quit the game at the preceding nodes. The process is repeated all the way back to time zero..4 Randomized Stopping Strategies Rather than limiting plans to a binary choice, the agent could consider randomizing the stopping decision so that a plan at node (i, t) is a probability p i,t which represents the probability of continuing at that node. A stopping strategy associated with a particular starting node is now a collection of probabilities at all nodes which are reachable from the starting node. A given collection of continuation probabilities yields a prospect of gains and losses, which can then be evaluated using PT preferences via (). In classical optimal stopping problems in the expected utility framework at each instant of time agents face a choice between stopping and continuing. If there is no probability weighting then there is no incentive to randomize: the expected payoff is linear in the probabilities and if

13 the value from stopping is V S and the value from continuing is V C, then the payoff V (θ) from a randomized strategy involving continuing with probability θ is V (θ) = θv C + ( θ)v S max{v S, V C }. Hence there is an optimal strategy which is pure, and the ability to randomize brings no benefit to the agent. This result extends to any situation where there is no probability weighting. 9 However, if probabilities are reweighted then it is no longer the case that V (θ) = θv C +( θ)v S. Moreover the optimal strategy is no longer characterized by the set of nodes at which the agent continues to gamble, but rather is characterized by the optimal continuation probability at each node. In this setting, one can still distinguish the three type of agents as in Barberis () by amending the definitions in Section.3 to allow for plans which consist of continuation probabilities, rather than binary mappings. In this paper we focus on PT. However, the observation that randomization brings benefits applies to other models with probability weighting, for example, the rank-dependent utility (RDU) class of Quiggin (98) and Yaari (987)..5 A One-Period Model As a building block for the analysis of larger models, and to illustrate the ideas, in Appendix A we analyze deterministic and randomized strategies in a one-period binomial setting. We make minimal assumptions on v, namely (simple) loss aversion, concavity on gains and convexity on losses, and assume only that the probability weighting function is inverse S-shaped and satisfies w( ). Under these properties, randomization is not of value to the agent, that is, he follows the same strategy as he would without the opportunity to randomize. The situation becomes more interesting in multi-period models. 3 The Two-Period Model with Tversky and Kahneman PT In this section we give results using the Tversky and Kahneman (99) specification of the value and weighting function (see () and (3)) with α ± = α and w ± = w. We take T =, h = 9 Henderson () considers an optimal stopping problem for an agent with prospect theory preferences, but with no probability weighting, and determines the optimal stopping rules. Since there is no probability weighting in her setting the proposed strategy is optimal, even if randomized strategies are allowed. 3

14 and PT parameters (α, δ, k) for which α [, ], δ [.3, ] and loss aversion k [, ]. We are interested in whether the agent enters the gamble, and the form of his optimal strategy, and the extent to which the answers to these questions depend on his ability to randomize. 3. The Agent who can Pre-commit Consider first Figure. The left panels show the strategies followed by the agent in the case where he cannot randomize, for three different parameter combinations. An open circle at a node indicates a node at which it is optimal to continue gambling; a closed or solid circle indicates a node at which it is optimal to stop. We find three possible behaviors - the loss-exit strategy of gambling at time zero and continuing after a win (ie. at (,)), but stopping after a loss (ie. at (,)) [Panel (a)]; the gain-exit strategy of continuing at (,) and at (,) but stopping at (,) [Panel (c)] and the trivial strategy of always stopping ie. not entering the game [Panel (e)]. Note it is never optimal to always continue. The right hand panels show what happens if, for the same parameter sets, we consider an agent who can randomize. We find cases where the agent who can randomize gambles at time zero when the agent who cannot does not gamble. The probability of gambling can be significant in this case, for the parameters of Panels (e) and (f) it is.47 as opposed to zero. However, compare panels (a) and (b) ((c) and (d)), if the deterministic agent follows a loss-exit (gain-exit) strategy then his counterpart with the ability to randomize follows a strategy with similar characteristics. Now consider the results of Figure which describes the strategies in detail as parameters change. Panel (a) shows the parameter combinations (in loss aversion k and α, we fix the weighting parameter δ =.5) for which the two flavors of agent gamble at time zero, ie. enter the casino. The agent without the ability to randomize follows a gain-exit strategy for low values of loss aversion k and α (region bounded by the orange solid line) and a loss-exit strategy for low k and large α (region bounded by red solid line) and for large values of loss aversion k (the region above the solid lines) chooses never to gamble. Also plotted in Panel (a) by the markers is the boundary of the region where the agent with the ability to randomize chooses a non-zero probability of gambling. (Again, the region below the symbols is the parameter region where he follows a non-trivial strategy). Note that the parameter region where the agent who can randomize enters the gamble at time zero is larger than the corresponding region for the deterministic agent. In this sense, the ability to randomize leads to more gambling. 4

15 The lower left panel (c) shows the optimal probability that the agent gambles at time zero for the agent who can randomize. The two right panels (b) and (d) show the probability of gambling at time, after a win (Panel (b)) or after a loss (Panel (d)). In cases where the agent who cannot randomize follows a gain-exit strategy then the agent who can randomize follows a similar strategy - he gambles at time zero and again after a loss, but after a win he may gamble again, albeit with a small probability. In cases where the agent who cannot randomize follows a loss-exit strategy then the agent who can randomize also exits after a loss, but gambles at time zero with a probability typically less than one, and sometimes as low as.4, and also gambles after a win with a probability less than one. There are several features of PT which drive these results. Firstly, the presence of loss aversion means that symmetric bets, such as stopping at time in all situations, are unattractive. (Further, increasing loss aversion parameter k reduces the value of all strategies making stopping at time zero more likely.) Secondly, the fact he is risk seeking on losses and risk averse on gains means there is an incentive to follow a gain-exit strategy. Since v()/v() = v( )/v( ) = α this incentive is greatest when α is small. However the impact of probability weighting is to provide incentives in the opposite direction - probability weighting gives greater prominence to extreme events, encouraging long tailed distributions on gains and thin tailed distributions on losses, or equivalently, encouraging loss-exit strategies. When α is large and thus the impact of convexity is small, this factor dominates. What then is the impact of randomization? In this model randomization gives the agent further flexibility in the design of his terminal wealth. This makes it more likely that the agent gambles or enters the casino initially, see Panel (a) of Figure. For the agent who is not allowed to make use of randomization, the only values of w which are relevant are w(/4) and w(/). In particular, there are no events of very small probability and the shape of w near zero is irrelevant. In constrast, the agent who can randomize can design gambles of arbitrary probability. 3. The Naive Agent Results for the naive agent are presented in Figures 3 and 4 in the same format. At time zero, the naive agent selects the identical strategy to the precommiting agent. That is, what he plans to do is the same as the agent who can commit. (See the time zero node in all trees in Figures and 3 and the left hand panels of Figures and 4.) The interesting feature of the naive agent is 5

16 the fact that because he is unable to commit, he has the possibility of modifying his strategy at time. The naive agent re-evaluates his strategies at time, and by the earlier analysis of the one-period model, always chooses to gamble after a loss but stop after a win, ie. if he gambles at time zero, he follows a gain-exit strategy. As Barberis () observes, the striking feature is that when the probability weighting is the dominant effect, the naive agent without access to randomization plans at time zero to follow a loss-exit strategy, but actually follows a gain-exit strategy. For the agent with randomization, the results are similar. The impact of randomization is to modify the probability of entering the casino initially, but at time, this agent still follows a gain-exit strategy. 3.3 The Sophisticated Agent The results for the sophisticated agent are obtained by backward induction, see Figure 5. In comparison with the results for the agent with the ability to commit, we find in the region where the latter follows a gain-exit strategy the two agents behave similarly, but in the region where the latter follows a loss-exit strategy, the sophisticated agent without the ability to commit follows the trivial strategy. The agent knows at time zero that he plans to follow a strategy of gain-exit at time (if he does not follow the trivial strategy). Hence, in regions where the agent with pre-committment would choose to follow a loss-exit strategy (where loss aversion and convexity/concavity are small and the probability weighting dominates) the sophisticated agent knows that his future self will not follow this strategy and prefers not to gamble at time zero. 3.4 How much better does the Agent do with Randomization? In Figure 6 we graph the time-zero PT value (assuming optimal behavior at time ) for the agent who can randomize (red solid line) and for the agent who follows a deterministic loss-exit strategy (blue dashed line) against the initial probability of entering the game at time zero. The graph takes parameters T =, and α =.6, k =.85 and δ =.5. Note that for these parameter values the agent with commitment (but unable to randomize) chooses to stop immediately. His next best strategy is the loss-exit strategy. An agent unable to randomize can only choose between entering with probability one (the 6

17 rightmost endpoint of the blue dashed line) or not entering (the leftmost endpoint of the blue dashed line). The latter strategy has value zero, and is the optimal choice (and the difference between the optimal strategy and the next best strategy is.). If we allow the agent to randomize (but only at time zero) then he should choose to gamble at time zero with a probability of.4, attaining a PT value of.5. But, the randomizing agent can choose probabilities of entering the gamble at all nodes at time. Allowing for this possibility, and using the optimal probabilities of continuing we find that the agent would prefer to gamble at time zero rather than stop immediately, but would strictly prefer to use a randomized strategy at time zero. In fact he chooses to gamble with probability.5, yielding a value of., which is the same order of magnitude as the impact of the deterministic agent following the next-best strategy. In particular, the gains from randomization are significant. 3.5 A Larger Number of Time Periods In Figures 7 and 8 we extend our analysis to the case T = 5. We take parameters α =.75, k =.5, δ =.5. This puts the planned (or pre-committed) agent s behavior if randomization is not considered into the regime of a loss-exit strategy. The results show that randomized strategies outperform non-randomized strategies. Nonetheless, the most striking conclusion of the Barberis analysis is carried over to this context: the naive agent plans to use a strategy which is similar to a loss-exit strategy, but in fact follows a strategy which is similar to a gain-exit strategy. 4 Continuous Time In the next two sections we discuss PT in a dynamic, continuous time model and consider the impact of the introduction of randomized stopping in this model. Ebert and Strack (4) analyze a continuous time, continuous state extension of the discrete time, discrete-state Barberis model of casino gambling. In this model, returns from gambling are modeled as a Brownian motion. Ebert and Strack (4) obtain the drastic result that in continuous time models, naive agents always postpone their stopping decisions: a naive agent with PT preferences will gamble until the bitter end. Taken at face value, this never stopping We further note that our parameters are extremely close to the example in Figure 4 of Barberis () which took the same k and δ but slightly less risk aversion/seeking with α =.95. 7

18 result leads to unpalatable and unrealistic predictions from the theory and casts doubt on the usefulness of PT preferences in a dynamic context. The assumptions of Ebert and Strack (4) preclude the use of randomized strategies., They show that the naive agent can design a stopping rule for which the stopping time is strictly positive and which he evaluates as strictly preferable to the alternative of stopping immediately. Hence the naive agent always finds it optimal to continue. We find that if the naive agent is allowed to randomize his strategy, then the strategy proposed by Ebert and Strack (4) can be improved upon by a strategy which involves sometimes stopping immediately and sometimes continuing according to the outcome of some exogenous event. Then, the naive agent sometimes stops. Thus, the extreme prediction of Ebert and Strack (4) that naive PT agents never stop, with the concomitant implication that PT is not appropriate for stopping models in continuous time need not hold for naive agents who can randomize their strategy. 4. Analysis Ebert and Strack (4) show that for an extremely wide class of value and probability weighting functions, there is a stopping rule which is preferred to stopping immediately and has the property that the probability of stopping at the current level is zero. They conclude that therefore, the naive agent never stops. But, for a PT agent, unlike an EUT agent, the fact an agent prefers one strategy over another does not mean that he necessarily prefers the first strategy over any mixture of the two. Indeed, as we now show, in the set-up considered in the main body of Ebert and Strack (4), the agent prefers such a mixture. Thus if the naive agent can find a way of mixing over strategies (eg. randomization) then he prefers sometimes stopping over never stopping. We assume the reference level is zero, and that the initial value of the Brownian motion is In Ebert and Strack (4), stopping rules are required to be adapted to the natural filtration of the relevant process, which is trivial at time zero. Hence in their model the naive agent cannot follow a strategy which requires them to stop immediately with probability θ (, ). This property is then inherited by his future self. Note that the agent with the ability to commit can manufacture strategies which are equivalent to randomized strategies, at least at all times t >, by deciding whether to continue to gamble or not based on an event which depends on the price history over [, t]. See Appendix B for a discussion of the differences between the assumptions of Ebert and Strack (4) and this paper. 8

19 at the reference level. Following Ebert and Strack (4), suppose we have an agent with value function v such that v is continuous, v() =, v is increasing and v ( ) = ( + c)v (+) for < c <. 3 Later we will add that v is concave on gains and also exhibits simple loss aversion in the sense v(x) + v( x) <. Suppose for the probability weighting functions w ± there exists a ˆp (, /) such that 4,5 w + (ˆp) > ( + c)ˆp + cˆp, w ( ˆp) < ˆp + cˆp. (4) These assumptions on the value and weighting functions mean that Assumptions and of Ebert and Strack (4) are satisfied. In this section we take the cumulative wealth from gambling to be a Brownian motion. 6 The PT value (see Kothiyal et al ()) of a continuous random variable X is given by V (X) = w + (P(v(X) > y))dy R + w (P(v(X) < y))dy. R Note that this definition nests the one for discrete variables given in (). We consider a naive agent and first give a version of Ebert and Strack s argument that such an agent keeps gambling. Let ɛ be a positive constant which later we will treat as a parameter. Let b ɛ = ɛ( ˆp) and a ɛ = ɛˆp. Suppose the agent has zero initial wealth. One strategy open to the agent (whether 3 Under these assumptions the value function exhibits finite marginal loss aversion in the sense that v (+) < v ( ) <. Köbberling and Wakker (5) introduce this notion of loss aversion and show that it has the advantage of being scale independent. We have added the adjective marginal to distinguish the concept from simple loss aversion v(x) + v( x) <. 4 For the first part of the argument we only require ˆp (, ) (as in Ebert and Strack (4)) but later we will require ˆp < /. This is automatic for inverse S-shaped probability weighting functions with w(/) < /. Then w(p) < p for all p (/, ) whence (+c)p > p > w(p) and the first inequality in (4) cannot hold for p > /. 5 +cp The condition (4) is satisfied by the commonly used inverse S-shaped weighting functions of Tversky and Kahneman (99), Prelec (998), Goldstein and Einhorn (987) and the neo-additive weighting function (Wakker,, p8) for all parameter values. A sufficient condition for there to exist a ˆp such that (4) holds is w +(+) > + c, w ( ) > + c. This property says extremely unlikely gains are overweighted and extremely likely losses are underweighted, both by more than the loss aversion parameter, ie. probability weighting is stronger than loss aversion. 6 Ebert and Strack allow for drifting Brownian motion, but this is not important for the argument that follows. See Appendix D. 9

20 naive or able to precommit) is to gamble until the first time his wealth reaches b ɛ or a ɛ and to stop at this instant. 7 By construction, and by the martingale property of Brownian motion, the probability that the process hits b ɛ before a ɛ is ˆp. Then the value Hˆp (ɛ) of this strategy is: Hˆp (ɛ) = w + (ˆp)v(b ɛ ) + w ( ˆp)v(a ɛ ) = w + (ˆp)v(ɛ( ˆp)) + w ( ˆp)v( ɛˆp). Note that Hˆp () =. Now, writing H ˆp for the derivative with respect to ɛ, { } ( + c) H ˆp (+) = ( ˆp)w +(ˆp)v (+) ˆpw ( ˆp)v ( + c) ( ) > ˆp( ˆp) v (+) =. + cˆp + cˆp Hence there exists ˆɛ > for which Hˆp (ˆɛ) > and for this (ˆp, ˆɛ) the agent prefers to continue (run until wealth first hits bˆɛ or aˆɛ ) over stopping immediately. From this, Ebert and Strack (4) conclude that the naive agent never stops. The preference for this gamble over stopping, which typically occurs when ˆp and ˆɛ are small, is named skewness preference in the small. 8 But what if this naive agent can maximize over randomized strategies? Suppose now he stops gambling immediately with probability θ and otherwise gambles until his wealth reaches bˆɛ or aˆɛ. Fixing ˆp, ˆɛ, considering θ as a variable and writing Hˆp,ˆɛ (θ) as the value of the mixed strategy, Hˆp,ˆɛ (θ) = w + (θˆp)v(bˆɛ ) + w (θ( ˆp))v(aˆɛ ) = w + (θˆp)v(ˆɛ( ˆp)) + w (θ( ˆp))v( ˆɛˆp). Note that Hˆp,ˆɛ () = is the value from stopping immediately with probability and Hˆp,ˆɛ () >. Consider the derivative of H with respect to θ. Then θ Hˆp,ˆɛ(θ) = ˆpw +(θˆp)v(ˆɛ( ˆp)) + ( ˆp)w (θ( ˆp))v( ˆɛˆp) [ = ˆp( ˆp) w +(θˆp) v(ˆɛ( ˆp)) + w ˆp (θ( ˆp)) v( ˆɛˆp) ] ˆp Suppose that v is concave on x > and that v(x) + v( x) <. Then v( ˆɛˆp) ˆp < v(ˆɛˆp) ˆp v(ˆɛ( ˆp)) < ˆp 7 Of course, if he is naive he will not be able to commit his future self to this strategy. 8 Note that Azevedo and Gottlieb () show there is also skewness preference in the large for the power shaped value function of Kahneman and Tversky (979).