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Embrace the randomness

by Dr Alex S.L. Tse, Research Associate, Cambridge Centre for Finance and Cambridge Endowment for Research in Finance

Abstract chaotic background

Excerpt from the CBS sitcom “The Big Bang Theory”, S05 E04:

Leonard: Are we ready to order?
Sheldon: One moment. I’m conducting an experiment.
Howard: With Dungeons and Dragons dice?
Sheldon: Yes. From here on in, I’ve decided to make all trivial decisions with a throw of the dice, thus freeing up my mind to do what it does best, enlighten and amaze. Page 14, item seven.
Howard: So, what’s for dinner?
Sheldon: A side of corn succotash. Interesting……

It sounds insane to let a die decide your fate. But we all know that our beloved physicist Dr Sheldon Cooper is not crazy (his mother had him checked!) so there must be some wisdom behind. To a mainstream economist, adopting randomisation in a decision task seems to violate a fundamental economic principle – more is better. By surrendering to Tyche the goddess of chance, we are essentially forgoing the valuable option to make a choice.

A well-known situation where randomised strategies are relevant is the game-theoretic setup where strategic interactions among players matter. A right-footed striker has a better chance of scoring a goal if he kicks left. A pure strategy of kicking left may not work out well though because the goalie who understands the striker’s edge will simply dive left. The optimal decisions of the two players thus always involve mixing between kicking/blocking left, right and middle etc. However, a very puzzling phenomenon is that individuals may still exhibit preference for deliberate randomisation even when there is no strategic motive. An example is a recent experimental study (Agranov and Ortoleva, Journal of Political Economy, 2017) which documents that a sizable fraction of lab participants are willing to pay a fee to flip a virtual coin to determine the type of lotteries to be assigned to them.

While the psychology literature offers a number of explanations (such as omission bias) to justify randomised strategies, how can we understand deliberate randomisation from an economic perspective? The golden paradigm of decision making under risk is the expected utility criteria where a prospect is evaluated by the linear probability-weighted average of the utility value associated with each outcome. There is no incentive to randomise the decision as the linear expectation rule would guide an agent to pick the highest value option with 100% chance. However, when the agent’s preference deviates from linear expectation, a stochastic mixture of prospects can now be strictly better than the static decision of sticking to the highest value prospect (Henderson, Hobson and Tse, Journal of Economic Theory, 2017). Rank-dependent utility model and prospect theory, which are commonly used in the area of behavioural economics, are two notable non-expected utility frameworks under which randomised strategies are internally consistent with the agent’s preference structure.

Incorporation of non-linear probability weighting and randomised strategies leads to many potential economic implications. For example, consider a dynamic stopping task where an agent decides whether to sell an asset at each time point. In a classical expected utility setup, there is no incentive for the agent to randomise the decision between to stop and to continue. This implies the optimal trading strategy must be a threshold-rule where sale only occurs when the asset price first breaches a certain upper or lower level. In reality, investors do not necessarily adopt this kind of threshold strategy even in a well-controlled laboratory environment. For example, the asset price could have visited the same level multiple times before a participant decides to sell the asset (Strack and Viefers, SSRN working paper, 2014). While expected utility theory struggles to explain trading rules that go beyond the simple “stop-loss stop-gain” style order, non-linear expectation and randomisation provide a modelling foundation to justify more sophisticated investment strategies adopted by individuals in real life.