My take on higherorder game theory
This is how I currently think about higherorder game theory, the study of agents thinking about agents thinking about agents....
This post doesn’t add any new big ideas beyond what was already in the post by Diffractor linked above. I just have a slightly different perspective that emphasizes the “metathreat” approach and the role of nondeterminism.
This is a work in progress. There’s a bunch of technical work that must be done to make this rigorous. I’ll save the details for the last section.
Multiple levels of strategic thinking
Suppose you’re an agent with accurate beliefs about your opponents. 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. Your beliefs are accurate, although for now we’ll be vague about what exactly “accurate” means.
In a game of Chicken, you may want to play Swerve, since that’s the maximin strategy. This is zerothorder thinking because you don’t need to predict what your opponent will do.
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 firstorder thinking.
Or maybe you know your opponent will use firstorder thinking. So you resolve to go Straight, your opponent will predict this, and they will Swerve. This is secondorder thinking.
Beyond secondorder, Chicken turns into a commitment race.
In Prisoner’s Dilemma, zeroth and firstorder thinking recommend playing Defect, as that’s a dominant strategy. Secondorder 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. Let’s say it’s worth it to Cooperate with probability if your opponent thereby Cooperates with probability more than (although the exact numbers depend on the payoff matrix).
If you’re a thirdorder agent, you might commit to Cooperating with probability equal to 51% times the probability that your opponent Cooperates. This causes your opponent to Cooperate with certainty, and thus you’re bound to Cooperate with probability 51%. This is a Paretooptimal outcome that favors you heavily.
Fixed points
So far we’ve talked about agents of order playing against agents of order . What about games between agents of equal order?
If two firstorder agents play the best response to each other, then they play a Nash equilibrium. Let’s see how this works in Matching Pennies, which has a mixedstrategy Nash equilibrium:
If the Even player thinks Odd is more likely to play Heads, then Even will play Heads. If the Even player thinks Odd is more likely to play Tails, then Even will play Tails. If Even thinks Odd is equally likely to play Heads or Tails, then Even will be indifferent. So if both players have wellcalibrated beliefs, we must be in the equilibrium where both players are indifferent and both players play Heads, Tails.
(What if Even always plays Heads when it’s indifferent? In that case, there is no equilibrium, which is awkward for our theory. But this behavior requires Even to represent its beliefs as real numbers. If we make the more realistic assumption that Even’s beliefs are arbitraryprecision numbers, then it can never be sure that it’s on the decision boundary. It gets stuck in a for loop, querying its belief for ever more precision in case it’s not exactly . (Then some other process, like infinitesimal errors in the beliefs, causes the agent to output the equilibrium strategy of Heads, Tails.))
The space of all agents
Now we’re ready to define our types. (I’ll leave some details to the last section.) The space of 0thorder agents is just the set of actions or pure strategies.^{[1]}
The space of storder agents is the space of functions on probability distributions on . That is, .
So an storder agent is a stochastic function from its thorder beliefs to a successor agent of order . Creating a successor agent is a form of currying.
Playing a game
Two agents in interact as follows:

Find distributions on such that and .
So is a fixpoint representing consistent thorder beliefs that the agents have about each other.^{[2]}

Sample from and sample from .

Repeat steps 1 and 2 on and until we end up with in .

Calculate the payoff to the first player.
Now take the expectation over all samples in step 2, and take the minimum over all choices of fixpoint in step 1. This gives a lower bound on the first player’s expected payoff.
Some agents
Now we can write down formulas for the kind of higherorder reasoning discussed earlier.
The firstorder agent that always plays the best response to its opponent is defined by . It achieves a Nash equilibrium when playing against itself.
A secondorder agent that always plays the best response to its opponent is .^{[3]} This agent also achieves a Nash equilibrium when playing against itself. In a future post I’ll construct a different 2ndorder agent that both plays the best response to any firstorder agent, and also achieves a Paretooptimal outcome when playing against itself.
Summary
Order  Beliefs about other agent  Strategic behavior  n vs. n equilibrium

0  None  Maximin  Maximin

1  0th order  Best response  Nash equilibrium

2  1st order  Argmax/optimize  Pareto optimum

...
Infinity
There’s a sense in which the 0thorder maximin agent is an approximation of the 1storder bestresponse agent, which is itself an approximation of the 2ndorder argmaxing agent. Standard domaintheoretical constructions should allow construction of infiniteorder agents, including an infiniteorder strategic agent that subsumes all finiteorder bestresponse behavior. Fortunately the hierarchy ends at infinity; has type . However, whether and how this works depends a lot on the technical details I haven’t worked out, so I hesitate to say any more about it.
Correlated strategies
This framework doesn’t have correlated strategies built in. But I have a hunch that agents of sufficiently high order can play correlated strategies anyways.
Technical details
Everything is a domain; in particular, a dcpo. is also a lower semilattice. The mapping spaces are spaces of Scottcontinuous functions.
is some kind of probabilistic powerdomain. An approach similar to (Jones & Plotkin 1989) might just work: would be the set of probability measures on the Borel algebra of which restrict to continuous evaluations on open sets.
But it’s possible will have to contain sets of probability measures, in order to express nondeterministic choice of fixpoint. Possible approaches include (Mislove 2000), (Tix, Keimel, Plotkin 2009), (GoubaultLarrecq 2007), and of course (Appel and Kosoy 2021). There’s also the field of quantitative semantics, which I don’t know anything about.
I’d like to have fixpoints which are maximal elements of . This might follow easily from the Kakutani fixed point theorem, but it really depends on how is defined.
Another issue is that higherorder strategic agents have to compute fixpoints themselves (in order to calculate the expected value of playing a lowerorder policy). I’m not sure if this can be done continuously. It might require the use of a reflective oracle somehow.
This work was supported by a grant from the Center on LongTerm Risk. Thanks to Alex Appel, Adele DeweyLopez, and Patrick LaVictoire for discussions about these ideas.
Actually, we want the set of nonempty sets of actions, so an agent can express indifference between actions. ↩︎
Really we want and to be maximal elements of the poset such that and . See the section on technical details. ↩︎
Actually you can’t compute an argmax over a function of a continuous variable. The best you can do is a distribution that’s biased towards higherpayoff outcomes, like quantilization or bestof sampling. I’ll write more about this in the future. ↩︎
Tangential, but did you ever happen to read statistical physics of human cooperation?
No, I just took a look. The spin glass stuff looks interesting!
Are we talking about the same thing?
https://www.sciencedirect.com/science/article/am/pii/S0370157317301424
Yep, I skimmed it by looking at the colorful plots that look like Ising models and reading the captions. Those are always fun.
I have a question about this entirely divorced from practical considerations. Can we play silly ordinal games here?
If you assume that the other agent will take the infiniteorder policy, but then naively maximize your expected value rather than unrolling the whole gameplaying procedure, this is sort of like ω+1. So I guess my question is, if you take this kind of dumb agent (that still has to compute the infinite agent) as your baseline and then rebuild an infinite tower of agents (playing other agents of the same level) on top of it, does it reconverge to A∞ or does it converge to some weird Aω2?
I think you’re saying Aω+1:=[ΔAω→ΔA0], right? In that case, since A0 embeds into Aω, we’d have Aω+1 embedding into Aω. So not really a step up.
If you want to play ordinal games, you could drop the requirement that agents are computable / Scottcontinuous. Then you get the whole ordinal hierarchy. But then we aren’t guaranteed equilibria in games between agents of the same order.
I suppose you could have a hybrid approach: Order ω+1 is allowed to be discontinuous in its orderω beliefs, but higher orders have to be continuous? Maybe that would get you to ω2.