Game Theory #2

To illustrate another similar example from last post, now supposed ALL the A-level Candidates met up in the town square. Prior to meeting up, they were all told by the head examiner that if everyone gets the same grade, all of you get an A. Else, the highest will have an A while the rest will fail. One of the students then suggested during he meet up, that why then they all fail the exam together and get zero marks. That will guarantee everyone an A. Supposed you’re one of the candidates there, will you be an angel and follow or a devil that deviate and score 1 mark.

This is of a similar nature to what we saw previously regarding the Prisoner’s Dilemma, only difference is that we have a lot more integrities to consider. After all, there is no telling an angel from a devil, So will you be an angel or devil?

To properly solve this, we need to define a few more things.

A strategy s_{i} {\in} S_{i} is a strictly dominant strategy for player i in game G= [I, [S_i ], \{u_i( \bullet) \} ] if for all {s \prime}_i \neq s_i, we have u_i (s_i, s_{-i}) > u_i (s'_i, s_{-i}) for all s_{-i} \in S_{-i}

A strategy s_i \in S_i is a strictly dominated strategy for player i in game G = [I, \{ S_i \}, \{u_i(\dot) \}] if there exists another strategy s'_i \in S_i such that for all s_{-i} \in S_{-i}, we have u_i(s' \prime _i, s_{-i}) > u_i (s_i, s_{-i}).

Okay, so let us look at these definitions closely first, before you guys go like meh sets?! This is a definition, and definitions tend to be boring. Let me begin by saying that the index i refers to the player, 1 or 2 for instance. and u refers to utility (payoff) while s here represents the various strategies that the player players. Now that the variables are sort of defined, are you starting to see how this definition is making sense? And ‘ here refers to a complementary strategy.

Next, a decision maker is rational if the strategy she chooses is at least as good as any other available strategy. Clearly, a rational player maximises her (expected) payoff.

  1. In the prisoner”s dilemma we saw, rational prisoners will rat on each other.
  2. In the angel vs devil students’ dilemma we saw, rational students will still do their best.Of course, this two cases differ in the number of people participating, which brings us to the next focus: Iterated deletion of strictly dominated strategies.

Here, we assume that rationality is common knowledge, that is, all man is rational. Thus, as a decision, he will never play a strictly dominated strategy (since it does not maximise his payoff))

Before I go further, I should clarify that payoff is different from income. Consider a dictator allocating an amount of money between himself and another player, the dictator might be altruistic or inequality averse (since its based on his beliefs) and thus maximises her utility (payoff) differently, instead of income. And we remind ourselves again, that a rational player will want to maximise his utility.

So next time, we will look at iterated deletion of strictly dominated strategies.

Click here for Game Theory #1

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