In many democratic countries, the timing of elections is flexible. We explore this potentially valuable option using insights from option pricing in finance.
The paper offers three main contributions on this problem. First, we derive a rationally-based mean-reverting political support process for the parties, assuming that politically heterogeneous voters continuously learn over time about evolving party fortunes. We solve for the long-run density for this process and derive the polling process from it by adding polling noise.
Second, we explore optimal timing using the political support process. The incumbent sees its poll support, and must call an election within five years of the last election to maximize its expected total time in office. This resembles the optimal exercise rule for an American financial option. This option is recursive, and the waiting and stopping values subtly interact. We prove the existence of the optimal exercise rule in this setting, and show that the expected longevity is a convex-then concave function of the political support. Our model is tractable enough that we can analytically derive how the exercise rule responds to parametric shifts.
We calibrate our model to the Labour-Tory rivalry in the U.K., with polling data from 1943-2005 and the 16 elections after 1945. Excluding three elections essentially forced by weak governments, our maximizing story quite well explains when the elections were called, and beats simple linear regressions. We also measure the value of election options, finding that over the long run they should more than double the expected time in power of a fixed term electoral cycle.
JEL Classification No.: D83, D72, G1
Keywords: American option, European option, Brownian motion, Electoral timing
Consider Becker’s classic 1963 matching model, with unobserved fixed types and stochastic publicly observed output. If types are complementary, then matching is assortative in the known Bayesian posteriors (the ‘reputations’).
We discover a robust failure of Becker’s result in the simplest dynamic two type version of this world. Assortative matching is generally neither efficient nor an equilibrium for high discount factors. In a labor theoretic rationale, we show that assortative matching fails around the highest (lowest) reputation agents for ‘low-skill (high-skill) concealing’ technologies. We then find that as the number of production outcomes grows, almost all technologies are of either form.
Our theory implies the dynamic result that high-skill matches eventually break up. It also reveals that the induced information rents create discontinuities in the wage profile. This in turn produces life-cycle effects: young workers are paid less than their static marginal product, and old workers more.
Keywords: Assortative matching, Incomplete information, Wages, Bayesian posterior, Value function
We introduce and solve a new class of “downward-recursive” static portfolio choice problems. An individual simultaneously chooses among ranked stochastic options, and each choice is costly. In the motivational application, just one may be exercised from those that succeed. This often emerges in practice, such as when a student applies to many colleges.
We show that a greedy algorithm finds the optimal set. The optimal choices are “less aggressive” than the sequentially optimal ones, but “more aggressive” than the best singletons. The optimal set in general contains gaps. We provide a comparative static on the chosen set.
Keywords: College application, Submodular optimization, Greedy algorithm, Directed search
There are two varieties of timing games in economics: In a war of attrition, more predecessors helps; in a pre-emption game, more predecessors hurts. In this paper, we introduce and explore a spanning class with rank-order payoffs that subsumes both as special cases. In this environment with unobserved actions and complete information, there are endogenously-timed phase transition moments. We identify equilibria with a rich enough structure to capture a wide array of economic and social timing phenomena — shifting between phases of smooth and explosive entry.
We introduce a tractable general theory of this class of timing games based on potential functions. This not only yields existence by construction, but also affords rapid characterization results. We then flesh out the simple economics of phase transitions: Anticipation of later timing games influences current play — swelling pre-emptive atoms and truncating wars of attrition. We also bound the number of phase transitions as well as the number of symmetric Nash equilibria. Finally, we compute the payoff and duration of each equilibrium, which we uniformly bound. We contrast all results with those of the standard war of attrition.
Keywords: Timing game, War of attrition, Pre-emption game, Potential function, Nash equilibrium
We study infinitely repeated games with observable actions, where players have present-biased (so-called beta-delta) preferences. We give a two-step procedure to characterize Strotz–Pollak equilibrium payoffs: compute the continuation payoff set using recursive techniques, and then use this set to characterize the equilibrium payoff set U(beta,delta). While Strotz–Pollak equilibrium and subgame perfection differ here, the generated paths and payoffs nonetheless coincide.
We then explore the cost of the present-time bias. Fixing the total present value of 1 util flow, lower beta or higher delta shrinks the payoff set. Surprisingly, unless the minimax outcome is a Nash equilibrium of the stage game, the equilibrium payoff set U(beta,delta) is not separately monotonic in beta or delta. While U(beta,delta) is contained in payoff set of a standard repeated game with smaller discount factor, the present-time bias precludes any lower bound on U(beta,delta) that would easily generalize the beta = 1 folk-theorem.
This paper produces a comprehensive theory of the value of Bayesian information and its static demand. Our key insight is to assume ‘natural units’ corresponding to the sample size of conditionally i.i.d. signals – focusing on the smooth nearby model of the precision of an observation of a Brownian motion with uncertain drift. In a two state world, this produces the heat equation from physics, and leads to a tractable theory. We derive explicit formulas that harmonize the known small and large sample properties of information, and reveal some fundamental properties of demand: (a) Value ‘non-concavity’: The marginal value of information is initially zero; (b) The marginal value is convex/rising, concave/peaking, then convex/falling; (c) ‘Lumpiness’: As prices rise, demand suddenly chokes off (drops to 0); (d) The minimum information costs on average exceed 2.5% of the payoff stakes; (e) Information demand is hill-shaped in beliefs, highest when most uncertain; (f) Information demand is initially elastic at interior beliefs; (g) Demand elasticity is globally falling in price, and approaches 0 as prices vanish; and (h) The marginal value vanishes exponentially fast in price, yielding log demand. Our results are exact for the Brownian case, and approximately true for weak discrete informative signals. We prove this with a new Bayesian approximation result.
Keywords: Value of information, Non-concavity, Heat equation, Demand, Bayesian analysis
This paper revisits Wald’s (1947) sequential experimentation paradigm, now assuming that an impatient decision maker can run variable-size experiments each period at some increasing and strictly convex cost before finally choosing an irreversible action. We translate this natural discrete time experimentation story into a tractable control of variance for a continuous time diffusion. Here we robustly characterize the optimal experimentation level: It is rising in the confidence about the project outcome, and for not very convex cost functions, the random process of experimentation levels has a positive drift over time. We also explore several parametric shifts unique to our framework. Among them, we discover what is arguably an `anti-folk’ result: Where the experimentation level is positive, it is often higher for amore impatient decision maker.
This paper more generally suggests that a long-sought economic paradigm that delivers a sensible law of demand for information is our dynamic one — namely, allowing the decision maker an eternal repurchase (resample) option.