We can model the behavior of different actors in the Bitcoin network for a given situation using a game theoretical analysis. Our first assumption is that actors will act in a rational way, where rationality is defined as taking the actions that maximize their utility. In our scenario, utility is defined by the monetary gains resulting from an action. While not everyone in Bitcoin is purely motivated by monetary gain, this is still a powerful generalization.
A Pure Strategy Nash Equilibrium is the set of actions that maximize each actor’s utility given the responses of all the other actors. Note, the utility each player receives depends on the player’s own decisions and the decisions of all the other actors, in the same way that the rewards a miner gets in the mining pool depend on the miner’s own decision to attack or cooperate and the rest of the pool’s decision to attack or cooperate. Players in the Bitcoin network or in a mining pool will converge to acting according to the Pure Strategy Nash Equilibrium for a given scenario, assuming they all behave rationally.
Let’s take a look at a simple scenario. Suppose there are only 2 mining pools in the Bitcoin network, Pool A and Pool B. Their utilities are shown below. The numerical value of utility is arbitrary in economics; it does not have any associated units, and it is calculated using a utility function of other parameters, defined by the economist. We are only interested in the comparative value of an action’s utility, whether it is higher or lower than another; we are not interested in its absolute value. In this example, the respective utilities are derived from the monetary gains each player would receive from the given scenarios of attacking or cooperating with the other.
The specific numerical values for the utilities in this example were chosen arbitrarily to reflect a scenario where players are incentivized to act dishonestly, but fare worse when both players are dishonest than when both players are honest. Let’s take the perspective of Player A. Given Player B acts honestly, player A gains the most utility from acting dishonestly, since a utility of 3 is greater than the utility of 2 that A would receive by acting honestly.
Given Player B acts dishonestly, Player A would prefer also acting dishonestly since a utility of 1 >utility of 0.
Player B makes the same conclusions, and makes preferences as follows:
We can see that the scenario in which both players choose to act dishonestly is the Pure Strategy Nash Equilibrium, since at this position, both players are maximizing their payoffs given the other player’s actions. Thus, despite receiving higher returns from both acting honestly, Players A and B will both act dishonestly.
If you are interested in learning more about introductory game theory, we recommend the following reading: A Brief Introduction to Basics of Game Theory Opens in new window by Matthew O. Jackson, Stanford University.