Option games

Option games integrate game theory with real options and represent a translation of the basic characteristics of game theory within a real option framework (Azevedo and Paxson 2010). In this respect the fundamental element of game theory is the players themselves. In a real option context, ”players” represent the companies facing a real investment option. For example, the option to develop an R&D project for a new molecule. Based on competitive conditions as well as development outcomes, a firm can decide whether to anticipate or defer the project, expand the project in a new indication or stop it. All these outcomes represent the ”strategy” of a company playing a game on the market when faced with uncertain dynamics regarding the demands and pressures that exist through competition. When playing a strategy, each player‘s goal is to reach certain objective results or outcomes which are quantified in numerical terms as payoffs. Payoffs are a means of representing the utility achieved with a certain strategy. Given the uncertainty surrounding the probability of reaching these outcomes, the reference is to expected payoffs. The corresponding terminology for ”payoffs” in real option terms is a company’s “value function”. This represents the payoff a firm achieves by playing a spe- cific strategy for any value assumed by the underlying stochastic variables. This dependence on stochastic variables makes the value function rather complex, while the payoff function in standard games are generally represented by simpler, more deterministic values. In the exam- ple previously mentioned in section 2.2.1 regarding the two friends deciding on whether to attend the theatre, or the stadium; the payoffs of each player are represented by deterministic

values that represent the utility gained for each player when playing each specific strategy. The difference between deterministic payoffs and value functions has implications in terms of how the equilibrium solutions are defined, as will be shown later.

At this point it is important to highlight that in these situations, uncertainty refers not only to the possibility of obtaining a certain amount of payoff but also to the time taken for its realization, and this will be next explored.

Option games in continuum time

In a real option game setting, each firm needs to not only account for its own optimal strategy, but also for a competitor‘s optimal option exercise strategy. In game terms, this means that the equilibrium strategies are a company‘s Best Response (Dias and Teixeira 2010).

Firms face uncertainties in an investment opportunity and uncertainty is modeled by fluctuations in the stochastic variable. Firms are assumed to use Markovian strategies leading to Markovian equilibria that depend only on the current state of the stochastic variable. This is because the history at each period can be summarized by the current state of the of the stochastic variable. The current state determines the current payoffs levels (Fudenberg and Tirole 1985).

Option games in continuous form are particularly concerned with the solution of optimal stopping time; that is the identification of the specific strategy that represents an optimal option exercise (Ziegler 2004).

Analogously to pricing perpetual American options, where the firm has no deadline on when to invest, the solution is in continuum and allows for the identification of the threshold x∗ which separates the waiting (continuation) area from the stopping area. The continuation area is representative of a waiting strategy interrupted at the threshold x∗, signaling the beginning of the stopping region. The threshold x∗ triggers the option exercise thus stopping to wait. The reference is to the first hitting time (or stopping time): the firm invests at the time t∗, the infimum time when the stochastic process X first reaches the value x∗ approaching this level from below (Jeanblanc, Yor, Chesney 2009).

The optimal time t∗ at which the first player stops is defined in Fudenberg and Tirole (1985) as

t∗= min{t|at_i =stop for at least one i}

payoff and the competitor gets the follower payoff.

Timing issues affect all of a firm’s strategies, such as choosing an optimal time to invest so as to to gain the competitive advantage by being the first to patent and preempting others from entering the market. Similarly, the player who is second to the market must decide how much time to dedicate to the development of an improved version of the leader’s drug. These are optimal decisions that pharmaceutical firms must make so as to maximize the value of their options.

Option games in discrete time

Option games in discrete time form value R&D opportunities using a binomial tree. In this situation the underlying uncertain variable is analyzed using a two stage game model that considers both upward and downward movements (Smit and Ankum 1993). Basic insights into the investment timing strategy in duopoly are obtained by analyzing SPE and applying backward induction along the tree. Project values, outcomes or payoffs are developed along the tree and are represented in deterministic values (as in standard games).

The level of information available at each decisional node reflects an important element of the game. For example, a situation of perfect information is defined by the knowledge of historical and current details at each node and has different implications than a game of imper- fect information. A game of perfect information assumes no uncertainties, which translates in graphical terms in informations sets consisting of single nodes. In this case there is no ambigu- ity about what has happened. Conversely when information sets include two or more nodes the game is of imperfect information and one of the players does not know what has occurred.