Dominance and the Prisoner's Dilemma

A cartel that cannot hold itself together

In 2009 the South African Competition Commission opened one of the continent's most consequential antitrust files: the country's largest cement producers had, for years, coordinated prices and quietly carved up regional markets. What is striking is not that the arrangement existed, but that it kept breaking down and had to be renegotiated. Firms with every incentive to keep prices high nonetheless found themselves shaving a little off to win one contract, then another, until the collusive edifice cracked. Why would rational managers, fully aware that discipline was collectively profitable, behave in a way that destroyed the very profits they were protecting?

The answer is not weakness of character. It is the logical structure of the situation itself — a structure so common that it has its own name, the Prisoner's Dilemma, and so sharp that the outcome can be read off from three ideas: dominance, the elimination of strategies no rational player would ever use, and the gap between what is individually rational and what is collectively best. This module builds those tools and turns them on cartels, tax evasion, grazing commons, and trade policy across Africa.

Dominance and iterated elimination

Strict and weak dominance

Fix a strategic-form game: a set of players, a set of strategies for each, and a payoff to every player for each combination of choices. You pick a strategy, your rivals pick theirs, payoffs follow. Before asking what others will do, ask a sharper question: is one of your strategies a good idea no matter what they do? If a strategy of yours yields a strictly higher payoff than another against every possible profile of opponents' choices, the second strategy is worthless — you can discard it without knowing anything about how your opponents reason.

The strict-versus-weak distinction matters more than it first appears. Discarding a strictly dominated strategy is uncontroversial: no belief about your opponents could ever justify it. Discarding a weakly dominated one is subtler, because there may be opponent choices against which it does exactly as well — it is only strictly worse in some scenarios and never better, so a cautious player might still keep it as insurance. Hold on to this asymmetry; it returns to bite us the moment we start eliminating strategies in sequence.

Iterated elimination and its limits

If you would never play a strictly dominated strategy, and you know your rival is equally rational, then you know your rival will never play one either — so you may delete their dominated strategies too. That deletion can expose a strategy of yours that was not dominated in the original game but is dominated in the reduced one. Delete it, look again, repeat. This is iterated elimination of strictly dominated strategies (IESDS). It formalises the idea that rationality is common knowledge: each player is rational, knows the others are, knows that they know, and so on up the ladder.

To watch that last sentence break, consider the game below. Each cell lists (your payoff, your rival's payoff); you choose the row, your rival the column. Your strategy M weakly dominates both of your other rows, and column L weakly dominates column R for your rival. Nothing is strictly dominated to begin with, so every elimination we make is a weak one — and the order will now matter.

            L        R
T           1, 1     0, 0
M           1, 1     2, 1
B           0, 2     2, 1

First delete T, which M weakly dominates. In the two-row game that remains, your rival finds R weakly dominated by L, so R goes; then your B is strictly worse than M, so it goes too, leaving the single profile (M, L). Now rewind and instead delete B first, which M also weakly dominates. Your rival again drops R, but the remaining rows T and M yield identical payoffs against L — neither dominates the other, so both survive. Two legitimate orders, two different answers: deleting a weakly dominated strategy can erase an outcome that ought to have survived.

The Prisoner's Dilemma

The dilemma was constructed in 1950 by Merrill Flood and Melvin Dresher at RAND and given its enduring parable by Albert Tucker: two suspects held separately, each offered a deal to inform on the other. But the structure has nothing to do with prisons. Give two players a choice between Cooperate and Defect, and label the four outcomes by their payoff to a single player: T for the temptation of defecting while the other cooperates, R for the reward of mutual cooperation, P for the punishment of mutual defection, and S for the sucker's payoff from cooperating while the other defects.

Read that ordering slowly, because everything else follows from it. Whatever your opponent does, defecting pays you more: if they cooperate you earn T rather than R; if they defect you earn P rather than S. So a rational player defects — no forecast of the opponent is even needed, since defection is dominant. When both reason this way they land on (P, P), the one cell each would gladly trade for (R, R). The tragedy is not that the players are foolish; it is that they are individually rational.

Worked example: an African cement cartel

Return to the cement producers and strip the market to two firms — call them Firm A and Firm B, though the logic scales to the handful of large producers that dominate most national cement markets in Africa, from Nigeria to Kenya to South Africa. Cement is close to a commodity — buyers care mainly about price, which makes quiet price-cutting both tempting and hard to detect in time. Each firm chooses either to Hold the agreed collusive price or to secretly Undercut it. Suppose profits, in illustrative units, are as follows.

                    B: Hold        B: Undercut
A: Hold             10, 10          2, 14
A: Undercut         14,  2          5,  5

Solve it by dominance, from Firm A's seat. Suppose B Holds (left column): A earns 14 by Undercutting versus 10 by Holding, better by 4. Suppose instead B Undercuts (right column): A earns 5 by Undercutting versus 2 by Holding, better by 3. Undercut beats Hold in both columns, so Undercut strictly dominates Hold for A — and by the symmetry of the matrix, for B too. The four numbers obey 14 > 10 > 5 > 2, precisely the ordering T > R > P > S. The unique dominant-strategy equilibrium is (Undercut, Undercut), paying each firm 5, even though both would earn 10 under (Hold, Hold). The cartel collapses into a price war it could see coming.

Two caveats keep this honest. First, the sharp prediction is a property of the one-shot game. Cement firms meet in the market month after month, and when a dilemma is repeated indefinitely the threat of future price wars can sustain cooperation — the subject of a later module on repeated games and the Folk Theorem. Second, real payoffs are estimates and detection is imperfect; the matrix is a lens for reasoning, not a spreadsheet. What survives both caveats is the core insight: absent some enforcement mechanism, the dominant-strategy logic pulls relentlessly toward defection.

From two firms to a whole society: n-player social dilemmas

The dilemma needs no more than the ordering, not two players. In an n-player social dilemma each person chooses to Contribute or Free-ride; free-riding yields a private gain regardless of what others do, yet if everyone free-rides all are worse off than if all had contributed. Defection dominates for each individual, and the collectively preferred outcome is not an equilibrium. The same three forces — a private temptation to defect, a collective reward for cooperation, a punishing outcome when all defect — reappear, now stretched across thousands or millions of people.

• Tax compliance. Public services are funded whether or not you personally pay, so each taxpayer is tempted to under-declare: the free-rider's private gain. If enough evade, the revenue base collapses and everyone endures worse roads, clinics and schools. Revenue authorities such as Kenya's KRA, Nigeria's FIRS and the South African Revenue Service are, in game-theoretic terms, the enforcement technology that shifts each citizen's dominant strategy back toward paying.
• Overgrazing the commons. On communally held pastoral land across the Sahel and East Africa, each herder gains privately from adding one more animal, but the pasture's carrying capacity is shared; when all follow the same dominant logic, the rangeland degrades for everyone. This commons dilemma is taken up in detail in a later module.
• Public goods generally. Vaccination, contributing to a savings group, maintaining an irrigation channel, honouring a fishing quota — each shares the structure of individually rational defection undermining a collectively better outcome.

Here is the lesson that recurs for the rest of the course. Individual rationality and collective welfare are not the same thing, and in a social dilemma they point in opposite directions. Markets, which usually convert self-interest into shared gain, offer no such guarantee here; left alone, the dominant strategy is defection all the way down. Whatever bends the outcome back toward cooperation — repetition, reputation, enforceable contracts, well-designed institutions, or the tax collector — has to be deliberately built. The rest of game theory is, in large part, the study of how.

Exercises

                       B: Free Trade   B: Tariff
A: Free Trade          4, 4            1, 5
A: Tariff              5, 1            2, 2

            L        C        R
U           3, 6     2, 3     1, 2
M           4, 5     3, 4     2, 1
D           1, 2     0, 1     0, 0