Lab 2: Cartels in the Lab
Textbook reading: none. This lab needs no preparation.
Key Ideas
Lab 1’s games were played once. Today the same players meet again and again, and every choice is revealed before the next round. A player’s choice today can now depend on what the other did yesterday, and both players know it, so a player can make threats and promises about what they will do next. The question of the lab is whether that changes what a points-only player does, and whether it changes what a room does.
Two ways of playing again are worth separating. In one, everyone knows exactly how many rounds there are, so the last round can be seen coming. In the other, a die decides after every round whether there is another. The third game replaces partners with rivals: four sellers of the same product, each choosing a high or a low price, round after round, with nothing to stop anyone from cutting.
As in Lab 1, you will predict on the whiteboard for forty minutes and play for ten. For each game, predict what points-only players do and what this room will do, and where the two come apart, say why.
Game 1: Five rounds. Two players play cooperate or fink (both cooperate: 5 each, both fink: 0 each, one finks: finker 8, cooperator −3), played five times with the same partner, choices revealed after every round. Everyone knows it is exactly five rounds.
Suppose one player says: “I will cooperate as long as you cooperate. If you ever fink, I will fink for the rest of the game.” Is the promise to cooperate credible? Start with Round 5 and work backward to predict what game theory says should happen in all five rounds. Then predict what you think people in this class will actually do (be specific: what portion of students will pick C and F in each round?). Will behavior match the theory? Why or why not?
Game 2: Until the die says stop. Same game, same partner. After every round the instructor rolls a die: 1 to 5, play another round; 6, the match is over. Is the promise from game 1 credible in this case?
Game 3: Four sellers. Groups of four, one product. Each round every firm prices high or low, revealed after the round. The die ends the match as in game 2.
| Others choosing low | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| You choose high | 10 | 2 | 0 | 0 |
| You choose low | 20 | 8 | 5 | 3 |
What’s the one-shot Nash Equilibrium? What if players get one minute to talk before each round: is the promise credible to choose high as long as everyone else has chosen high? What if players have to play in silence, and the instructor reports only how many players chose low, not which ones?
Solutions
Theory predicts that players will fink in every round. Start with Round 5, which is just a one-shot Prisoner’s Dilemma, so both players fink. Knowing that Round 5 will end in finking, cooperation in Round 4 cannot buy future cooperation, so both fink in Round 4 as well. The promise to cooperate is never credible. The same logic unravels back to Round 1.
In practice, students will often cooperate early and fink more frequently as Round 5 approaches. There is no single correct behavioral prediction, but a reasonable prediction might be something like 70% C in Round 1, 65% in Round 2, 55% in Round 3, 40% in Round 4, and 20% in Round 5.
Until the die says stop. Now there is no known final round, so backward induction cannot begin. After every round there is a (5/6) probability of another round.
If both players follow the strategy “cooperate until someone finks, then fink forever,” cooperating gives an expected payoff of \(5 + \frac{5}{6} (5) + \frac{5}{6}^2 (5) + \frac{5}{6}^3 (5) + ...\), which converges to 30.
Finking while the other player cooperates gives 8 immediately, followed by 0 in future rounds if the punishment is carried out. Since 30 > 8, neither player wants to fink. Thus cooperation can be sustained as an equilibrium.
Low beats high whatever the others do, so the one-shot prediction is all low, 3 each, when all high gives 10. Played in silence, the high price often holds for a few rounds and collapses after the first cut. Talk usually produces a clear deal and longer cooperation, though nothing said is binding. Hiding who cut removes the ability to punish the right player: a player that hears “one chose low” cannot tell whom to punish, punishing everyone destroys the arrangement, and letting it slide invites more cutting.