Excel Models
The Game Theory & Real Options
Game theory is a fascinating and complex discipline that requires years of study to master. Here, I’ll cover the main directions and strategic insights, and finally, upload Excel models to illustrate these concepts.
Key Strategic Games
We can categorize strategic games in the following ways:
- Games with simultaneous decisions
- Games with unfavorable equilibria
- Zero-sum or constant-sum games
- Coordination games
- Games with sequential decisions
- Infinite games (both simultaneous and sequential)
Simultaneous Games with Unfavorable Equilibria
These include the well-known “prisoner’s dilemma,” where two isolated participants decide independently whether to confess to a crime. Here, a dominant strategy leads both players to an equilibrium state, where both confess, resulting in an 8-year sentence each, though with coordination, they could have received only 2 years each. The question “What if I don’t confess, but my partner does?” mathematically drives both sides toward confession.
This scenario applies to oligopoly games, the Cold War (arms spending), marketing wars, and political conflicts, where cooperation would yield better outcomes, but dominant strategies create a temptation to defect.
Zero-sum or Constant-sum Games
Many of us have played zero-sum games like “odds and evens” or strategized about where to kick a ball to score a goal. In such games, the total “pie” is fixed: one player’s gain is the other’s loss. Here, there’s no dominant strategy, requiring players to change strategies each round (to prevent the goalkeeper from predicting their moves). Optimal strategies, such as kicking 50% left and 50% right, are based on probabilities (e.g., the goalkeeper’s strengths). In more complex cases, like penalty kicks, a 2/5 left and 3/5 right split may be better, based on skill levels. The goal is to create proportions where outcomes remain statistically the same, regardless of opponent decisions.
Complex version:
A product is being developed (let’s say a chip) that has two opposing characteristics: graphics and speed. Improving the graphics decreases speed, and vice versa. In this case, graphics are slightly more important to the market.
The leading company prefers that the follower cannot differentiate its product, so it can maintain its lead. Therefore, it must make investment decisions that prevent the follower from knowing in advance whether graphics or speed will be prioritized. Conversely, the follower benefits from differentiating its product. The payoffs are distributed as follows:
If their strategic decisions align, profits are split 5/4 in favor of graphics; if the strategies do not align, profits are distributed 6/2 in favor of the leader.
According to the formulas, in each subsequent step, the leader will invest in graphics with a probability of 3/5 and in speed with a probability of 2/5 (the formulas are available in the Excel sheet).
Coordination Games
In coordination games, several equilibria may exist. For instance, two competing companies might need to agree on a technology standard (e.g., HDMI or USB). Without coordination, both receive lower returns than if they coordinate, though one may benefit more depending on the chosen standard. Thus, despite differing benefits, coordination is necessary to avoid losing altogether.
Sequential Games
These differ from simultaneous games by allowing players to make decisions one after another. An example is market entry, where one decides based on a competitor’s likely response. If entering the market triggers intense competition, both may lose; however, if adapting is better for the competitor, entry becomes favorable. This is where “commitment” plays a role—if a competitor blocks its own options to prevent backing down, a fierce response becomes more likely, deterring market entry.
But what happens if, before I make my entry decision, the competitor manages to block their own “not fight” option by committing? In that case, a fierce reaction would be less costly for them than adapting. And since I anticipate that the competitor will respond aggressively, entering the market is no longer a rational choice for me.
Infinite Games
Though parallel or sequential games may span 100 years, they can often be broken down into sub-games, allowing the discovery of dominant strategies. However, if a game is truly endless, equilibria need not hold. Known as “folk theorem,” this idea posits that certain players’ behaviors, like persistently fierce competition, may discourage new entrants without needing extra deterrents.
Game Theory and Real Options
Real options analysis may be limited in scenarios where competition is a factor, beyond technical or market risks. For instance, delaying an investment to observe macroeconomic trends might seem prudent, but in monopolistic or oligopolistic markets, it could lead to losing market share to competitors. Thus, in such cases, decisions should be assessed through a game theory lens. There is a sample from the book*
In conclusion, game theory provides a structured approach to strategic decisions in competitive environments. Examples discussed are illustrated in the Excel file: The Game Theory.
Source: Venture Capital & the Finance of Innovation by A. Metrick & A. Yasuda