![]() But this behavior requires Even to represent its beliefs as real numbers. (What if Even always plays Heads when it's indifferent? In that case, there is no equilibrium, which is awkward for our theory. So if both players have well-calibrated beliefs, we must be in the equilibrium where both players are indifferent and both players play 1 2 Heads, 1 2 Tails. If Even thinks Odd is equally likely to play Heads or Tails, then Even will be indifferent. If the Even player thinks Odd is more likely to play Tails, then Even will play Tails. If the Even player thinks Odd is more likely to play Heads, then Even will play Heads. Let's see how this works in Matching Pennies, which has a mixed-strategy Nash equilibrium: ![]() If two first-order agents play the best response to each other, then they play a Nash equilibrium. What about games between agents of equal order? So far we've talked about agents of order n + 1 playing against agents of order n. This is a Pareto-optimal outcome that favors you heavily. This causes your opponent to Cooperate with certainty, and thus you're bound to Cooperate with probability 51%. If you're a third-order agent, you might commit to Cooperating with probability equal to 51% times the probability that your opponent Cooperates. ![]() Let's say it's worth it to Cooperate with probability p if your opponent thereby Cooperates with probability more than p 2 (although the exact numbers depend on the payoff matrix). Second-order thinking says that it's good to Cooperate on the margin if by doing so you cause your opponent to also Cooperate on the margin. In Prisoner's Dilemma, zeroth- and first-order thinking recommend playing Defect, as that's a dominant strategy. This is second-order thinking.īeyond second-order, Chicken turns into a commitment race. So you resolve to go Straight, your opponent will predict this, and they will Swerve. Or maybe you know your opponent will use first-order thinking. Or maybe you predict what your opponent will do and play the best response to that, Swerving if they'll go Straight and going Straight if they'll Swerve. This is zeroth-order thinking because you don't need to predict what your opponent will do. In a game of Chicken, you may want to play Swerve, since that's the maximin strategy. Your beliefs are accurate, although for now we'll be vague about what exactly "accurate" means. It doesn't matter where your beliefs come from perhaps you have experience with these opponents, or perhaps you read your opponents' source code and thought about it. Suppose you're an agent with accurate beliefs about your opponents. I'll save the details for the last section. There's a bunch of technical work that must be done to make this rigorous. ![]() I just have a slightly different perspective that emphasizes the "metathreat" approach and the role of nondeterminism. This post doesn't add any new big ideas beyond what was already in the post by Diffractor linked above. This is how I currently think about higher-order game theory, the study of agents thinking about agents thinking about agents. ![]()
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