Experiment 15 Correlated Equilibrium
Can Advice Help Everyone?
Compare separate random choices with private advice from a shared mediator, then test whether following is worthwhile.
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Start with “Work It Out, Step by Step” below. The optional expert section explains its symbols as you go. For more examples, use the plain-language math guide.
1. Make a Prediction
Would You Follow a Private Recommendation?
You and a partner must choose wait or go. Going alone pays well, but going together pays nothing. A mediator, a shared advice giver, draws a pair of recommendations. Each player sees only their own. The mediator cannot force either player to obey.
| Your choice | Partner waits | Partner goes |
|---|---|---|
| Wait | You 4; partner 4 | You 1; partner 5 |
| Go | You 5; partner 1 | You 0; partner 0 |
2. Make Your Choice
3. Reveal the Incentives
Check the Advice You Actually Received
Conditional means “given what you know.” If you are told to wait, look only at draws in which your recommendation is wait. Among those draws, compare following with changing. Do the same for every recommendation to each player. The partner is assumed to follow their own advice.
| Recommendation | Chance sent | Follow | Change | Incentive |
|---|
Independent choices are drawn separately: your draw gives no clue about the partner’s. Correlated recommendations are drawn together, so your advice can carry a clue about theirs. A correlated equilibrium is a shared drawing rule for which neither player gains on average by changing after seeing their own advice, assuming the other follows. A tie is allowed.
4. Change One Assumption
Put Different Advice Pairs in the Bag
Think of each weight as a count of matching tickets in a bag. A weight of 2 gives that pair twice the chance of a pair with weight 1. Zero means the pair is never drawn. The app divides each weight by the total to calculate its probability, or chance.
Try all four weights at 1: that is the same as two separate fair coin flips. Then make only “both wait” positive. It looks good for the group, but each player could improve by going instead. Compare that temptation with the starting weights of 1, 1, 1, and 0.
Changing a weight or seed resets the drawing sequence and your prediction. A seed is a number that lets you repeat the same random draws. All-zero weights are invalid because the bag would be empty. Calculated averages assume both follow; your personal trial lets you depart from that assumption.
Work It Out, Step by Step
Use the starting bag: one “both wait” ticket, one “you wait, partner goes” ticket, and one “you go, partner waits” ticket. There is no “both go” ticket.
- Suppose you are told to wait. Only two tickets could explain your advice. On one, the partner waits; on the other, the partner goes. Each is equally likely.
- Compare your choices. Following wait gives 4 or 1, averaging \((4+1)/2=2.5\). Changing to go gives 5 or 0, averaging \((5+0)/2=2.5\). It is a tie, so you do not gain by changing on average.
- Suppose you are told to go. Only one ticket fits. The partner must have been told to wait. Following earns 5. Changing earns 4. Following is better.
- Check the partner too. The bag treats the two players the same way, so their checks give the same answers. No recommendation tempts either player to improve by changing. That makes this bag a correlated equilibrium.
- Add up the benefit. Following all three possible tickets gives you 4, 1, and 5 points. Your average is \((4+1+5)/3\), about 3.33. The partner averages the same. Together, that is about 6.67 points per draw.
If each player instead chooses go independently one-third of the time, both go together one-ninth of the time. The separate chances match the starting advice bag, but the pairing differs. The combined average is about 6.22. The advice improves that comparison by changing which choices occur together. It does not make every possible mediator better than every independent arrangement.
For experts: formal model and assumptions
Joint and Conditional Probabilities
- \(p_{WW}\), \(p_{WG}\), \(p_{GW}\), and \(p_{GG}\)In words: “probability of wait-wait,” “wait-go,” “go-wait,” and “go-go.”
- W abbreviates wait and G abbreviates go. The first lower letter is your recommendation; the second is the partner’s. These are joint probabilities, the chances that a pair occurs together. Each is at least zero, and all four add to 1. The lower letters label a pair; they do not indicate multiplication.
- \(P(B=G\mid A=W)\)In words: “the probability that B is told go, given that A is told wait.”
- A names your recommendation and B names the partner’s. P means probability. The upright bar means “given,” not division. See conditional probability.
In words: “Among all the draws where you are told to wait, find the fraction where the partner is told to go.” The denominator, the number below the fraction line, is the chance you are told to wait. If it is zero, that advice is never sent and this conditional probability is not defined. Its obedience check is then unnecessary.
The Four Obedience Checks
Obedience means following the recommendation. A deviation is choosing the other action. For each recommendation, multiply the score from following minus the score from changing by the chance of each matching pair, then add. A zero or positive result means following is at least as good.
In words: “For each of the four advice cases, the weighted gain from following must be greater than or equal to zero.” The sign \(\geq\) means “greater than or equal to.” For your wait advice, following loses 1 compared with changing when both were told wait, but gains 1 when the partner was told go. That produces the first inequality, a comparison rather than an equality.
When a recommendation has positive probability, dividing its weighted gain by that probability gives its conditional follow-minus-change average. Division by a positive number preserves whether a value is negative, zero, or positive, so the four comparisons above are equivalent to the conditional checks in the table. Never-sent recommendations have weighted gain zero.
The independent comparison uses each player’s marginal probability, the chance of their own action after adding over the other player’s possible actions. Your go chance is \(p_{GW}+p_{GG}\); the partner’s is \(p_{WG}+p_{GG}\). Independent pair chances multiply those separate chances. Thus it preserves individual action frequencies while removing their correlation. These independent choices are not necessarily an equilibrium. Separate 50% go probabilities do form a mixed Nash equilibrium in this particular payoff table: each player chooses randomly and neither can improve its average by changing only its own choice probabilities.
This is a single-round game with two players and two choices each, known points, and a known joint recommendation rule. Each player observes only their own recommendation before choosing. The app samples private advice but computes incentive comparisons exactly. There is no communication, extra payment between players, or enforcement after advice is drawn. The partner follows by assumption, including under invalid advice rules; those cases demonstrate a temptation rather than predict rational partner behavior. Ties count as obedience-compatible, with a tiny numerical tolerance for computer rounding.
Concept reference: Algorithmic Game Theory, Chapter 2, on correlated equilibria and the drivers game.
5. Take It Somewhere Else
Would Teams Follow a Shared Schedule?
A scheduler privately recommends different times for two teams to use a shared machine. The schedule avoids conflicts when both follow it. Is avoiding conflicts enough to make the advice voluntary?
Reveal the reasoning
No. Each team must prefer following, or at least be equally happy following, after seeing its own recommendation and using what that reveals about the other team’s schedule. A schedule can have a high combined payoff while giving someone a reason to ignore it.
A Model Is a Place to Start
These small models make the incentives visible. Their results follow from their stated rules; they are not forecasts of how every person or organization behaves. A simulated strategy is a rule, not a personality.
Scenario links save the controls and random seed. To reproduce an interactive run, make the same choices in the same order. Changing a setting restarts the experiment.