The Prisoner's Dilemma EXplained
Liam Scotchmer
All references are quoted and listed below.
4 Minute Read
Summary
Game theory predicts likely outcomes, the Nash equilibrium for >2 players based on their preferences.
The Prisoner’s Dilemma is the oldest, most simplest game of game theory but it shows how each individual’s decision is rational yet the collective outcome is not Pareto optimal. *Cooperation is possible, and is most optimal.
When there is a situation with more than two players, and there are known payouts or quantifiable consequences, we can use game theory to determine the most likely and optimal outcome
Again, Game theory does not force players to have certain preferences (as critics claim). Instead, game theory analyses what should happen given what players desire. (Spaniel)
Applications of the Prisoner’s Dilemma includes tariffs, war, arms race, or advertising. (Spaniel)
Whilst the outcome of prisoner’s dilemma is not Pareto optimal, in a deadlock the outcome is Pareto optimal: the players’ payoffs can not be increased. No option like cooperation exists.
The following is a summary chapter 1; Game Theory 101 - The Complete Textbook (Spaniel).
“The Prisoner’s dilemma is the oldest and most studied model in game theory and its solution concept is also the simplest.” (Spaniel) Nevertheless, it is so important because it shows how the equilibrium is stable and each individual’s decision is rational yet creates an outcome that collectively is suboptimal.
The story:
Two prisoners are caught just before robbing a jewellery store, and are arrested for trespassing. However the police suspect they planned to break in, but are unable to accuse them for such.
The pair are being interrogated individually. The interrogator offers each “player”, let’s say player 1 and 2:
1) to confess / cooperate, and in return, receive no punishment and partner will receive full punishment of 12 months
2) to keep quiet
However if both confess, each individual's testimony will no longer be as valuable, and the jail sentence will be 8 months each.
This sounds confusing, so let's turn this information into a “matrix”:
The Prisoner’s Dilemma
The first player’s strategies are in the rows, and the second player’s strategies are in the columns.
Eg.: if player 1 stays quiet, and player 2 confesses; then the game ends in the top right with a set of payoffs of <-12,0>.
Which strategy should each player choose?
To find out, let’s focus in.
Assuming player 2 stays quiet (so we’re in the “quiet” column), what should player 1 choose? Well, firstly, player 1 doesn’t care about player 2’s playoffs, so I have replaced them with question marks.
0 months is better than 1 month of jail (-1), so the optimal play for player 1 is to confess here (so, 0).
Assuming player 2 confesses, what should player 1 do? -8 is better than -12, so confessing is again, the optimal play here since 8 months is less than 12 months of prison time.
Now let’s switch to player 2’s perspective.
Assuming player 1 stays quiet, confessing here is better for player 2 as 0 months of jail time is better than 1!
Assuming player 1 confesses, the optimal play is for player 2 to confess, again!
Thus, the best decision for both players is to confess! This is called the Nash equilibrium. The justice system has worked, and both will spend 8 months in jail. You might be perplexed by this and ask why the players didn’t stay quiet, as <quiet,quiet> is better than <confess,confess>, but staying quiet “is inherently unstable.” (Spaniel). Confessing has the highest payoff in both choices. “Ultimately, the players finish in the inferior (but sustainable) <confess, confess> outcome.” (Spaniel)
When there is a situation with more than two players, and there are known payouts or quantifiable consequences, we can use game theory to determine the most likely and optimal outcome. (Heyes, 2024) However, typically, and important to note: the most preferred outcome has the highest score, compared to the least preferred outcome with the lowest score. (Spaniel)
For example, in this game, 12 could have been replaced with 1, 8 with 2, 1 with 3, and 0 with 4 -> so the higher the score the better. Even with those changes, confess is still the better option. (Spaniel)
The Meaning Behind The Numbers
Game theory does not force players to have certain preferences (as critics claim). Instead, game theory analyses what should happen given what players desire. (Spaniel)
Tariffs are a good example of the prisoner’s dilemma.
The reasoning behind the outcomes:
The best outcome (=4) is for one country to have tariffs on incoming goods, with no tariff on their goods being exported.
Free trade is the next best outcome, because nobody is losing (3).
Mutual tariffs are next because there’s a domestic advantage, but at the same time, a disadvantage abroad. (2)
The worst is to not enforce tariffs whilst the other country enforces tariffs. (1)
Let’s turn this information into a matrix:
This article argues the matrix looks different.
Uses of the Prisoner’s Dilemma
Quickly:
Country 1 perspective:
1. When player 2 chooses no tariff: Tariff (4) > 3 No tariff (3) = tariff is best option
2. When player 2 chooses tariff: Tariff (2) > No tariff (1) = tariff is the best option
Tariff is the best option under both of player 2’s decisions.
Country 2 perspective
1. When player 1 chooses no tariff: tariff (4) > no tariff (3)
2. When player 1 chooses tariff: tariff (2) > no tariff (1)
For both country’s, placing a tariff on the other country is the best outcome. Placing tariffs strictly dominates not placing tariffs in this setup.
Deadlock - Contrasting Prisoner’s Dilemma
In the prisoner’s dilemma, the equilibrium is not Pareto optimal: implying that the players’ payoffs can all be increased if they cooperate. (“Pareto Optimal - Game Theory .Net”)
However, in a deadlock, each player has a dominant strategy and the equilibrium is Pareto optimal, meaning the players’ payoffs can not be increased. (Shor) This means the strictly dominant strategy is also the most beneficial one. (Picardo)
This is shown below, where defect dominates in every case.
(Table courtesy of (Shor))
Thus, both players pursue the strictly dominant strategy (defection for both), which is the best collective result. No alternative like mutual cooperation Pareto dominates it (unless one player incorrectly makes their choice). (Spaniel)
Game theory predicts likely outcomes, the Nash equilibrium, for >2 players based on their preferences, even if the collective outcome is suboptimal (prisoner’s dilemma).
Playing a strictly dominated strategy is irrational.
The prisoner's dilemma is underpins advanced game theory strategies.
Close
References
“An Introduction to Game Theory.” Gocardless.com, 2021, gocardless.com/en-au/guides/posts/game-theory-explained/.
Brad Spangler. “Positive-Sum / Zero-Sum / Negative-Sum Situations.” Beyond Intractability, 7 July 2016, www.beyondintractability.org/essay/sum.
Eleftherakou, Olga. “Game Theory - an Introduction Game Theory Is a Branch of Mathematics and Economics That Studies Strategic Interactions between Rational Decision-Makers. It Analyzes Situations Where the Outcome of One Person’s Decisions Depends on the Decisions of Others.” Linkedin.com, 7 June 2024, www.linkedin.com/pulse/game-theory-simply-briefly-explained-olga-eleftherakou-wql1f/.
Hayes, Adam. “Game Theory: A Comprehensive Guide.” Investopedia, 27 June 2024, www.investopedia.com/terms/g/gametheory.asp.
“Here’s How Game Theory Helps Explain Donald Trump’s Strategies : Networks Course Blog for INFO 2040/CS 2850/Econ 2040/SOC 2090.” Cornell.edu, 2016, blogs.cornell.edu/info2040/2016/09/19/heres-how-game-theory-helps-explain-donald-trumps-strategies/. Accessed 4 Feb. 2026.
Ordouard, Victor. “Tariffs, Trade, and the Prisoner’s Dilemma : Networks Course Blog for INFO 2040/CS 2850/Econ 2040/SOC 2090.” Cornell.edu, 2017, blogs.cornell.edu/info2040/2018/09/12/tariffs-trade-and-the-prisoners-dilemma/.
“Pareto Optimal - Game Theory .net.” Www.gametheory.net, www.gametheory.net/dictionary/ParetoOptimal.html.
Picardo, Elvis. “How Game Theory Strategy Improves Decision Making.” Investopedia, 2019, www.investopedia.com/articles/investing/111113/advanced-game-theory-strategies-decisionmaking.asp.
Shor, Mike. “Deadlock - Game Theory .net.” Gametheory.net, 2026, www.gametheory.net/dictionary/Games/Deadlock.html. Accessed 4 Feb. 2026.
Spaniel, William. Game Theory 101 : The Complete Textbook. U.St., Createspace, 2015.
Wikipedia Contributors. “Prisoner’s Dilemma.” Wikipedia, Wikimedia Foundation, 1 Apr. 2019, en.wikipedia.org/wiki/Prisoner%27s_dilemma.
Wikipedia Contributors. “Deadlock (Game Theory).” Wikipedia, Wikimedia Foundation, 17 Oct. 2024, en.wikipedia.org/wiki/Deadlock_(game_theory).
“Zero- vs Positive-Sum - Explanation and Examples.” Conceptually, conceptually.org/concepts/zero-vs-positive-sum.