Mixed Strategies

Here’s a simple question about game theory.

Imagine that you think (a) that coherent credences need not satisfy countable additivity, and you think (b) that sometimes mixed strategies are optimal, and so we should consider all mixed strategies in deciding what an agent should do. I’m not particularly fond of either (a) or (b), but I know that I have readers who endorse each, and I suspect that I have readers who endorse the conjunction of the two.

Now imagine we’re trying to analyse a game where a player has a countable infinity of option in front of them. Should we take one of the player’s options to be the mixed strategy where the player has, for each option, probability 0 of taking that option? If so, then (a) how do we evaluate how good a strategy that is, and (b) are there any cases where this is the optimal strategy?

Here’s a more particular instance of this puzzle. Row and Column are playing a game of infinite matching pennies. Each player has to select a positive integer. Row wins if they select the same number, Column wins if they select a different number. I imagine that the opponent of countable additivity will want to say that the solution to the game is that both players play the strategy of selecting a random number, with each number having probability zero of being selected. But I don’t know how the opponent of countable additivity figures out what the expected return of this strategy (or any other infinitary strategy) is for each player, so I don’t know why they would think it is an equilibrium. So what should someone who believes (a) and (b) say about this game?


I’m a bit late to the party on this one, but I wanted to note a very exciting service that David Bourget and David Chalmers have set up, PhilPapers. Here is how the site describes itself.

PhilPapers is a comprehensive directory of online philosophy articles and books by academic philosophers. We monitor journals in many areas of philosophy, as well as archives and personal pages.

At first glance, the page seems to do everything I wanted an online philosophy archive and index to do. The only downside is that I have so much reading to do now!


What is it to forget that p? A simple analysis is that S forgets that p iff at one time S knows that p, and at a later time, S does not know that p. But this can’t be right, for the following four reasons.

If at t1, S knows that p, and at t2, S is dead, so knows nothing, S has not forgotten that p.

At t1, George knows that he has hands. At t2, he reads the Meditations, and starts to doubt that he has hands. That is, he no longer believes he has hands. He doesn’t any more know that he has hands, but nor has he forgotten that he has hands.

At t1, John knows that the New Deal substantially lowered unemployment. At t2, John reads a newspaper column, in a usually reliable newspaper, saying that this was not true. With his new evidence, he now (quite reasonably) doubts that the New Deal substantially lowered unemployment. So he does not know this. But nor has he forgotten it.

At t1, Paul knows that his meeting is scheduled for 2pm. At t2, he gets an email (falsely) saying that the meeting has been moved to 3pm. Paul glances at the email, but doesn’t take in what it says. So he still believes that the meeting is at 2pm. Nevertheless, he doesn’t know this, for the email is a defeater for his knowledge. But nor has he forgotten the time of the meeting.

That seems to dispose of the simple theory fairly conclusively. But is there anything we can put in its place?

Linguistic Anarchy!

I’m one of those people who can often get over an inability to settle down to work by going out to a cafe.  Since I’m in Berkeley now, naturally the cafe I found this afternoon was no ordinary Seattle’s Best, but the Mediterraneum Caffé (Caffé Med) on Telegraph, former haunt of Ginsberg and other Beats, and the place that claims to have invented the latte.  

I asked for a small latte.  The young server paused and said, “would a medium be ok?”  I said “er, sure…” and she said “because technically if it’s in a cup smaller than this one (holding up a cup that would make a perfectly respectable soup bowl) then it’s not called a latte.  Actually, if it’s like a latte but in this cup (holding up a cup that is still generous for a coffee cup) it’s called a macchiato.“   

Having been influenced by old <a href=“http://itre.cis.upenn.edu/~myl/languagelog/archives/001677.html”>Language Log</a> <a href=“http://itre.cis.upenn.edu/~myl/languagelog/archives/000933.html”>posts</a> on Starbucks’ (you don’t say small you say tall) and Microsoft’s (Microsoft has no genitive) amateurish attempts to regiment language in various ways, I’m never very impressed by this sort of thing.  It’s not that I’m opposed to the regimentation of language in general—-in fact, I usually follow one of my old teachers in recommending that my logic students refrain from using valid in informal senses (valid point of view, valid claim etc.) and reserve the word for its technical senses (which are tricky enough as it is, given that many books reserve one technical use of the word for first order logical truths, as well as allowing the more well-known use on which it is a property of arguments or argument schemata in general.)   So anyway, that sentence got away from me.  It’s not that I’m opposed to the regimentation of language in general, but just that I reject the authority of just about everyone in imposing it, including Starbucks, but also including funky historical local coffee shops. 

So what’s the difference between what they’re doing, and what I feel justified in doing in my classes?  Well, I think it’s just that I have a good justification for the regimentation.  Reserving valid for the technical uses aids communication and understanding of the subject at hand.  A regimentation that makes it impossible to request a coffee like a medium latte, but smaller, by saying “small latte’‘ does not.  In fact, it seems like a snobbish attempt to wield power for the sake of it.  Similarly for the Microsoft and Starbucks examples.  

Am I right?  I can imagine someone defending the Starbucks example by claiming that the justification for having special names for their coffee sizes is artistic.  They want their  customers to have the best, most enjoyable most interesting/mysterious/exotic coffee-drinking experience possible, and what better justification could there be for their decision to name their sizes as they have? But even if that is so, it could only justify their introduction of the new expressions, not the outlawing of the old—-and hence not the regimentation.  

Anyway, though I wasn’t impressed by the no-such-thing-as-a-small-latte claim, neither am I impressed by people who are rude to young service workers, so I tried to make conversation, dredging up some faint memories about what a macchiato actually was:  “That’s interesting.  I thought a macchiato was where you just marked the expresso with foam?”  “Oh no,’‘ she said, “a machiatto is just like a latte but with less milk.”  And I just shut up and smiled and handed over my 4 bucks.   

Maybe Berkeley cafes are going to be more distracting than the ones in St Louis. 

And I’m back. Hello!

I’ve been so quiet around here for so long that you’ve probably stopped wondering what my name is doing in the panel on the top left.  But no more.  By invoking the magic words pre-tenure sabbatical I have found myself (more or less) settled at the University of California, Berkeley, with no teaching duties. It’s the beginning of the semester, Branden Fitelson and John MacFarlane are both teaching great looking seminars (though I’m going to be a little bit cautious about blogging their contents – not everyone wants what-I-said-in-seminar-today broadcast to the world) and it turns out that Berkeley serves coffee and cookies in the break during their colloquia. So the stars are pretty much all aligned. Stay tuned…

One More Conference

The call for papers for this year’s Bellingham Summer Philosophy Conference has just gone out, though I don’t believe there’s a link yet. But papers (of up to 25 pages) for consideration should be sent to Dennis Whitcomb (Dennis.Whitcomb-at-wwu-dot-edu) by March 1.


This post is just to bring attention to two upcoming conferences.

Firstly, the Arche Methodology Conference, taking place in St Andrews in April.  Daniel and I will both be giving papers, and they have a strong lineup of others.  Attendance for non-participants is apparently ‘limited’ so it’s probably worth registering early!

Secondly, Finn Spicer just emailed me about his conference Minds Brains and Beyond, taking place in Bristol in March, with several big names on the programme.  Pop over to the conference website for more info.

Sleeping Beauty Variations and Explanations

I’ve been thinking a bit about Sleeping Beauty, and I’ve found it a little easier to think about this variation on the original case. I was wondering whether anyone thinks this changes the case substantially.

On Sunday, Sleeping Beauty is told about the game setup, and a coin is tossed, but Beauty isn’t told the results.

On Monday, Beauty is woken iff the coin lands tails, is put back to sleep, and has her memory erased.

On Tuesday, Beauty is woken, stays awake for as long as she would have stayed awake for if she had been woken on Monday, then is put back to sleep.

On Wednesday, Beauty is woken, is told that it is Wednesday, and goes on with her life.

And the interesting question is, on each day, what should be her credence that the coin landed heads?

The primary change from the standard form of the story is to make the only day of waking if heads to be Tuesday, not Monday. This makes it somewhat easier to think about what Beauty should think on Wednesday. I also made it explicit that she doesn’t have her memory erased on Tuesday, and that she’s told on Wednesday that it is Wednesday.

The reason I’ve been thinking about this version of the case is that it makes it easier to see what Beauty should say on Wednesday. And the reason I was thinking about that is that I think it pulls apart two ways of thinking about the problem. (This is all inspired by Robert Stalnaker’s book Our Knowledge of the Internal World. But I don’t say that Stalnaker would endorse any of this.)

One way of thinking about the problem is in terms of centered worlds propositions. Beauty’s knowledge, at any moment, consists of the centered worlds that for all she knows are her centered world at that time. So on Tuesday, she can think the thought “This is Tuesday”, and this means that the centre is on Tuesday. And that’s a thought that is true on Tuesday and false on Wednesday.

Another way of thinking about the problem uses regular possible worlds propositions, but makes free use of demonstrative reference to times to ‘latch on’ to propositions about the time. So on Tuesday Beauty can think the thought “This1 is Tuesday.” She might not know whether that is true, but she can consider the proposition. And she can think that thought on Wednesday, if she has sufficient memory to track that demonstrated time. And that proposition doesn’t change its truth value over time.

Now here’s one nice consequence of the latter way of thinking about the puzzle. Let’s say we want to say that on Tuesday, her credence in heads should be 1/3. (I’m not endorsing this, but a lot of people do. And I’m silent here about (a) what we should say about Monday, or for that matter Sunday, or (b) what the dynamic explanation is of how we get from Sunday to Monday to Tuesday.) And let’s also say, as I think we really must, that when she learns it is Wednesday, her credence in heads should be 1/2. What should be the explanation of the change from Tuesday to Wednesday?

On the regular propositions approach, all that happens is that Beauty conditionalises on her new information. On Tuesday she thinks the thought “This1 is Tuesday.” She gives it credence 2/3, since she gives equal credence to each of the following possibilities.

  • This1 is Tuesday and the coin landed heads.
  • This1 is Tuesday and the coin landed tails.
  • This1 is Monday and the coin landed tails.

Then on Wednesday she learns that that1 was Tuesday, so she can drop the third possibility. Conditionalising on the falsehood of that possibility gives her a new credence in heads of 1/2. That seems like an elegant solution to one part of the Sleeping Beauty problem.


Here’s an argument that every respectable epistemologist will reject the conclusion of. But I wonder which premise most people will think is false.

  1. If S has a justified, true belief that p, then S is aware that p.
  2. If S is aware that p, then S knows that p
  3. So, if S has a justified, true belief that p, then S knows that p.

At a pinch, I’d say premise 2 is false. But I’d be interested to know which premise other people think is false.