Question

Consider the following situation. A Superior Being (SB) and human (P) are involved in an interaction. SB’s primary goal is for P to believe in its existence and its secondary goal is not to reveal itself. P’s primary goal is to decide based on evidence about SB’s existence or non-existence, while P’s secondary goal is to believe. Write down a payoff matrix consistent with these preferences, and solve the game using ideas from class.

Answer 1

This gives four possible outcomes, described in the following table:

		P
		Believe	Don't believe
SB	Reveal	P faithful with evidence	P unfaithful despite evidence
	Don't reveal	P faithful without evidence	P unfaithful without evidence

Let's assume that each player has a primary and a secondary goal. SB's primary goal is for P to believe in its existence and its secondary goal is not to reveal itself. P's primary goal is to have some evidence about SB's existence or non-existence, while P's secondary goal is to believe.

I assume that P is neither an avowed theist nor an avowed atheist but a person of scientific bent, who desires confirmation of either belief or non-belief. Preferring the former to the latter as a secondary goal, P is definitely not an inveterate skeptic. What SB might desire is harder to discern. Certainly the god of the Hebrew bible very much sought untrammeled faith and demonstration of it from his people. The justification for SB's secondary goal is that remaining inscrutable is SB's only way of testing its subjects' faith.

The goals listed above suggest a ranking of SB's outcomes: its favorite outcome is for P to believe without revelation (we'll give this a 4 on SB's preference scale), followed by P believing and SB revealing itself (a 3 on the scale), followed by P not believing and SB not revealing itself (a 2 on the scale), followed by P not believing and SB revealing itself (a 1 on the scale).

For P the ranking is: P believing and SB revealing itself (4), P not believing and SB not revealing itself (3), P believing and SB not revealing itself (2) and P not believing and SB revealing itself (1). We indicate the preferences by pairs of numbers, e.g (1,2), where the first number describes SB's ranking and the second one P's ranking:

		P
		Believe	Don't believe
SB	Reveal	P faithful with evidence (3,4)	P unfaithful despite evidence (1,1)
	Don't reveal	P faithful without evidence (4,2)	P unfaithful without evidence (2,3)

For SB it makes sense not to reveal itself: this strategy is better whether P selects belief (because SB prefers (4,2) to (3,4)) or non belief (because SB prefers (2,3) to (1,1)). For SB, not revealing itself is what is called a dominant strategy. Assuming that P knows SB's preferences, he can work out that non revelation is SB's dominant strategy. Not having a dominant strategy himself, but preferring (2,3) to (4,2), he will choose not to believe as the best response.

These strategies lead to the selection of (2,3): SB does not reveal itself and P does not believe. The outcome (2,3) is a Nash equilibrium of the game. It's considered an equilibrium because no player has an incentive to move away from it by itself.

Note, however, that (3,4) - P believes and SB reveals itself - is better for both players than (2,3). But it is not a Nash equilibrium because SB, once at (3,4), has an incentive to depart from (3,4) to (4,2). Neither is (4,2) an equilibrium, because once there, P would prefer to move to (2,3). And of course, both players prefer to move from (1,1).

Thus, if the players have complete information and choose their strategies independently of each other, game theory predicts the choice of (2,3): no revelation and no faith. Paradoxically, this is worse for both players than (3,4). This is also the paradox in game theory's most famous game, the prisoner's dilemma, although the details of the two games differ.