Johannes Treutlein

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Two Notions of Best Response

Wolfgang Spohn develops the concept of a "dependency equilibrium" based on a similar notion of evidential best response (Spohn 2007, 2010). A joint probability distribution is a dependency equilibrium if all actions of all players that have positive probability are evidential best responses. In case there are actions with zero probability, one evaluates a sequence of joint probability distributions such that and for all actions and . Using your notation of a probability matrix and a utility matrix, the expected utility of an action is then defined as the limit of the conditional expected utilities, (which is defined for all actions). Say is a probability matrix with only one zero column, . It seems that you can choose an arbitrary nonzero vector , to construct, e.g., a sequence of probability matrices The expected utilities in the limit for all other actions and the actions of the opponent shouldn't be influenced by this change. So you could choose as the standard vector where is an index such that . The expected utility of would then be . Hence, this definition of best response in case there are actions with zero probability probably coincides with yours (at least for actions with positive probability—Spohn is not concerned with the question of whether a zero probability action is a best response or not).

The whole thing becomes more complicated with several zero rows and columns, but I would think it should be possible to construct sequences of distributions which work in that case as well.

Smoking Lesion Steelman

Thanks for your answer! This "gain" approach seems quite similar to what Wedgwood (2013) has proposed as "Benchmark Theory", which behaves like CDT in cases with, but more like EDT in cases without causally dominant actions. My hunch would be that one might be able to construct a series of thought-experiments in which such a theory violates transitivity of preference, as demonstrated by Ahmed (2012).

I don't understand how you arrive at a gain of 0 for not smoking as a smoke-lover in my example. I would think the gain for not smoking is higher:

.

So as long as , the gain of not smoking is actually higher than that of smoking. For example, given prior probabilities of 0.5 for either state, the equilibrium probability of being a smoke-lover given not smoking will be 0.5 at most (in the case in which none of the smoke-lovers smoke).

Smoking Lesion Steelman

From my perspective, I don’t think it’s been adequately established that we should prefer updateless CDT to updateless EDT

I agree with this.

It would be nice to have an example which doesn’t arise from an obviously bad agent design, but I don’t have one.

I’d also be interested in finding such a problem.

I am not sure whether your smoking lesion steelman actually makes a decisive case against evidential decision theory. If an agent knows about their utility function on some level, but not on the epistemic level, then this can just as well be made into a counter-example to causal decision theory. For example, consider a decision problem with the following payoff matrix:

Smoke-lover:

  • Smokes:

    • Killed: 10
    • Not killed: -90
  • Doesn't smoke:

    • Killed: 0
    • Not killed: 0

Non-smoke-lover:

  • Smokes:

    • Killed: -100
    • Not killed: -100
  • Doesn't smoke:

    • Killed: 0
    • Not killed: 0

For some reason, the agent doesn’t care whether they live or die. Also, let’s say that smoking makes a smoke-lover happy, but afterwards, they get terribly sick and lose 100 utilons. So they would only smoke if they knew they were going to be killed afterwards. The non-smoke-lover doesn't want to smoke in any case.

Now, smoke-loving evidential decision theorists rightly choose smoking: they know that robots with a non-smoke-loving utility function would never have any reason to smoke, no matter which probabilities they assign. So if they end up smoking, then this means they are certainly smoke-lovers. It follows that they will be killed, and conditional on that state, smoking gives 10 more utility than not smoking.

Causal decision theory, on the other hand, seems to recommend a suboptimal action. Let be smoking, not smoking, being a smoke-lover, and being a non-smoke-lover. Moreover, say the prior probability is . Then, for a smoke-loving CDT bot, the expected utility of smoking is just

,

which is less then the certain utilons for . Assigning a credence of around to , a smoke-loving EDT bot calculates

,

which is higher than the expected utility of .

The reason CDT fails here doesn’t seem to lie in a mistaken causal structure. Also, I’m not sure whether the problem for EDT in the smoking lesion steelman is really that it can’t condition on all its inputs. If EDT can't condition on something, then EDT doesn't account for this information, but this doesn’t seem to be a problem per se.

In my opinion, the problem lies in an inconsistency in the expected utility equations. Smoke-loving EDT bots calculate the probability of being a non-smoke-lover, but then the utility they get is actually the one from being a smoke-lover. For this reason, they can get some "back-handed" information about their own utility function from their actions. The agents basically fail to condition two factors of the same product on the same knowledge.

Say we don't know our own utility function on an epistemic level. Ordinarily, we would calculate the expected utility of an action, both as smoke-lovers and as non-smoke-lovers, as follows:

,

where, if () is the utility function of a smoke-lover (non-smoke-lover), is equal to . In this case, we don't get any information about our utility function from our own action, and hence, no Newcomb-like problem arises.

I’m unsure whether there is any causal decision theory derivative that gets my case (or all other possible cases in this setting) right. It seems like as long as the agent isn't certain to be a smoke-lover from the start, there are still payoffs for which CDT would (wrongly) choose not to smoke.