by Michael Brunnbauer, 2013-08-22

The paper Bell nonlocality and Bayesian game theory contains much general math that is difficult for me to grasp because I am not familiar with Bayesian Games or Bell Inequalities.

But the last example game is concrete and can be calculated. Here is my calculation:

Two companies form a joint venture and bid for a piece of land rich with some resource. Company A knows about the extraction cost for this resource which can be low or high. Company B knows about the supply for this resource on the world market. Low costs or supply mean high profits while high costs or supply mean low profits. The companies are not allowed to exchange the information they have about the land before bidding and if they bid both, they get the land but will both have to pay for it so the profits will be lower. This is the payoff table for all situations:

 Company B Low supply High supply No bid Bid No bid Bid Company A Low cost No Bid 0 4 0 2 Bid 4 2 2 1 High cost No Bid 0 2 0 -2 Bid 2 1 -2 -2

If the players act independent, one optimal strategy is to bid if cost or supply is low, which yields an average payoff of 1.5 (2 + 2 + 2 + 0 / 4). The game rewards anticorrelated bidding (only one of the players bids) in 3 of 4 cases but the penalty in the one case equalizes the gains in the other 3 cases so that the average payoff stays at 1.5 (4 + 2 + 2 - 2 / 4).

In a universe with local realism - where correlations between distant entangled particles can be explained with hidden variables - no strategy yields a higher average payoff than 1.5. But in our universe, higher payoffs are possible using spin measurement of entangled particles in a singlet state without actually exchanging information (e.g. each company bids only when measuring spin-up). The key is this graphic from the Wikipedia article about Bell's theorem:

The graphic shows the correlation for spin measurement in a local realist universe (solid lines) and in our universe (cosine). A correlation of 1 means that both measurements yield the same spin, a correlation of 0 means that both measurements yield random spin and a correlation of -1 means that one measurements yields spin-up while the other yields spin-down.

The non-linearity of the cosinus allows to get a higher anticorrelation when rewarded without the same increase in anticorrelation when not rewarded:

Company A sets the measuring equipment to 0 degree if costs are low and to 60 degree when costs are high.

Company B sets the measuring equipment to 200 degree if supply is low and to 140 degree if supply is high.

Each company bids only when measuring spin-up.

Situation 1: low cost, low supply

Angle between detectors: 200 degrees, correlation = cos(200) = -0.9397
Payoff at correlation -1 (exactly one bids): 4
Payoff at correlation 0 (random bidding): 2.5
Payoff at correlation -0.9397: 3.90955 = 4 * 0.9397 + (1-0.9397) * 2.5

Situation 2+3: low cost, high supply or high cost, low supply

Angle between detectors 140 degrees, correlation = cos(140) = -0.7660
Payoff at correlation -1 (exactly one bids): 2
Payoff at correlation 0 (random bidding): 1.25
Payoff at correlation -0.7660: 1.8245 = 0.7660 * 2 + (1-0.7660) * 1.25

Situation 4: high cost, high supply

Angle between detectors 80 degrees, correlation = cos(80) = 0.1736
Payoff at correlation 1 (they bid together or none): -1
Payoff at correlation 0 (random bidding): -1.5
Payoff at correlation 0.1736: -1.4132 = 0.1736 * -1 + (1-0.1736) * -1.5

Average payoff: -1.4132 + 1.8245 + 1.8245 + 3.90955 / 4 = 6.14535 / 4 = 1.5363375

Here is a Python script that will determine optimum angles and payoffs for the classical and quantum case.

If I read the paper right, more general (hypothetical) quantum correlations allow the same average payoffs as if the players would in fact exchange information - without actual information exchange and violation of causality.