Game Theory: Prisoner’s Dilemma & Dominating Strategies
It is Monday, a sheer winter day of 2021. You sit soundless in your cell: cold, still and faint. All you can do is think of what you will miss in the year to come. For the crimes you and your friend will be convicted, it would seem a 365 day sentence is imminent. You chose to remain in silent opposition to the interrogation team. Surely your friend did as well…
A few moments later, you hear the news. Your close confidant is free. How is this possible?
It’s now Friday, December 5th. You walk out of the state penitentiary a wiser skeptic.
The year is 2031…
This brief story is one outcome of the famous Prisoner’s Dilemma in Game Theory. There are two criminals (players) in solitary confinement. They will be convicted of a lesser crime and possibly a greater one (provided one criminal testifies against the other). The number of years they serve in prison is dependent on their choices while in confinement.
Game outcomes:
- Both players betray each other, each get five years
- Both players cooperate with one another, each get one year
- Player 1 betrays while Player 2 cooperates, Player 1 is free and Player 2 gets ten years
- Player 2 betrays while Player 1 cooperates, Player 2 is free and Player 1 gets ten years
Often in mathematics it is helpful to compress outcomes into a table.
Figure 1 is called a normal form game. It is used to analyze games that are simultaneous, like our Prisoner’s Dilemma.
The inner four blocks are payoffs for each player in the form (Player 1, Player 2). For example, the top left block, (-1, -1), indicates a one year sentence for the players. Each player has a strategy set of two, giving way to four outcomes.
So, why did the friend choose to betray? This outcome doesn’t seem like something friends do to each other.
In Game Theory, we assume the agents are rational players whose objective function is to maximize payoff. Here, maximum payoff is least number of years in prison. The underlying assumption of rationality leads to interesting and effective ways of analyzing game.
Obviously, we don’t want to end up with the ten year sentence. A rational thing to do would be to avoid playing in such a manner to cause this outcome. Though we may be able to get more methodical with our approach…
Given that our players are rational beings, they assume this quality of rationality about each other. The cascading of assumptions about one another’s decision making gives way to the method of Iterated Dominance.
Let’s pretend we are Payer 1 for a moment. Looking at Player 2, we can see the following possible outcomes.
- Player 1 goes Silent, Player 2 gets 1 year for Silent or 0 years for Betray
- Player 1 goes Betray, Player 2 gets 10 years for Silent or 5 years for Betray
We can see that Player 2’s best interest is to betray us. Regardless of what we choose, the resulting payoff, also known as utility, for Player 2 is strictly greater when betraying.
In iterated dominance, we can simply remove the Silent strategy for Player 2. From Player 1’s perspective, Player 2 will not be silent. This reduction be be seen in Figure 2.
Likewise, we may eliminate the Silent strategy for Player 1, leading to a dominance solvable game. The resulting payoff mapping is referred to as the Dominating Strategy. Here, we are left with (-5, -5).
The definition for dominance in games is as follows:
Strict Dominance: strategy A strictly dominates strategy B if for any move by an opposing player the utility (payoff) of A is always greater than B.
Weak Dominance: strategy A weakly dominates strategy B if for any move by an opposing player the utility (payoff) of A is at least that of B, and for at least one move by the opposing player the utility (payoff) of A is greater than B.
In Prisoner’s Dilemma, the iterated dominance method follows strict dominance.
In our short story from earlier, the friend chose to play the situation in a game theoretic manner.
This model is just the beginning of Prisoner’s Dilemma analysis. For instance, how does the game change if it is played repeatedly? Check out the Prisoner’s Dilemma Wikipedia for more information on Iterated Prisoner’s Dilemma. You’ll find greedy decision making is not always the best choice!
Moreover, we did not have to use the numbers given in this example. In general, the game follows this rule:
For our greedy strategy of betrayal to hold, the variable payoffs must follow this rule:
X > Y > Z > W
In conclusion, Game theory is the study of how rational agents make decisions upon some interaction with one or more similar agents. Mathematical models are used to understand such strategic interactions, giving way to this field.
The axiom upon which all else builds is this pursuit of maximum payoff under some parameters of interaction, like strategy sets and number of games played.
Until next time!
