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Game Theory: Lecture 9 Transcript

October 3, 2007

Professor Ben Polak: So last time we saw this, we saw an example of a mixed strategy which was to play 1/3, 1/3, 1/3 in our rock, paper, scissors game. Today, we're going to be formal, we're going to define mixed strategies and we're going to talk about them, and it's going to take a while. So let's start with a formal definition: a mixed strategy (and I'll develop notation as I'm going along, so let me call it Pi, i being the person who's playing it) Pi is a randomization over i's pure strategies. So in particular, we're going to use the notation Pi (si) to be the probability that Player i plays si given that he's mixing using Pi. So Pi(si) is the probability that Pi assigns to the pure strategy si.

Let's immediately refer that back to our example. So for example, if I'm playing 1/3, 1/3, 1/3 in rock, paper, scissors then Pi is 1/3, 1/3, 1/3 and Pi of rock — so Pi(R) — is a 1/3. So without belaboring it, that's all I'm doing here, is developing some notation. Let's immediately encounter two things you might have questions about. So the first is, that in principle Pi(si) could be zero. Just because I'm playing a mixed strategy, it doesn't mean I have to involve all of my strategies. I could be playing a mixed strategy on two of my strategies and leave the other one with zero probability. So, for example, again in rock, paper, scissors, we could think of the strategy 1/2, 1/2, 0. In this strategy I assign — I play rock half the time, I play paper half the time, but I never play scissors.

So everyone understand that? And while we're here let's look at the other extreme. The probability assigned by my mixed strategy to a particular si could be one. It could be that I assign all of the probability to a particular strategy. What would we call a mixed strategy that assigns probability 1 to one of the pure strategies? What's a good name for that? That's a "pure strategy." So notice that we can think of pure strategies as the special case of a mixed strategy that assigns all the weight to a particular pure strategy. So, for example, if Pi(R) was 1, that's equivalent to saying that I'm playing the pure strategy rock, i.e. a pure strategy.

So there's nothing here. I'm just being a little bit nerdy about developing notation and making sure that everything is in place, and just to point out again, one consequence of this is we've now got our pure strategies embedded in our mixed strategies. When I've got a mixed strategy I really am including in those all of the pure strategies. So let's proceed. I'm going to push that up a little high, sorry. So now I want to think about what are the payoffs that I get from mixed strategies, and again, I'm going to go a little slowly because it's a little tricky at first and we'll get used to this, don't panic, we'll get used to this as we go on and as you see them in homework assignments and in class.

So let's talk about the payoffs from a mixed strategy. In particular, what we're going to worry about are expected payoffs. So the expected payoff of the mixed strategy P, let's be consistent and call it Pi, the mixed strategy Pi is what? It's the weighted average — it's a weighted average or a weighted mixture if you like — of the expected payoffs of each of the pure strategies in the mix. So this is a long way of saying something again which I think is a little bit obvious, but let me just say it again.

October 3, 2007

Professor Ben Polak: So last time we saw this, we saw an example of a mixed strategy which was to play 1/3, 1/3, 1/3 in our rock, paper, scissors game. Today, we're going to be formal, we're going to define mixed strategies and we're going to talk about them, and it's going to take a while. So let's start with a formal definition: a mixed strategy (and I'll develop notation as I'm going along, so let me call it Pi, i being the person who's playing it) Pi is a randomization over i's pure strategies. So in particular, we're going to use the notation Pi (si) to be the probability that Player i plays si given that he's mixing using Pi. So Pi(si) is the probability that Pi assigns to the pure strategy si.

Let's immediately refer that back to our example. So for example, if I'm playing 1/3, 1/3, 1/3 in rock, paper, scissors then Pi is 1/3, 1/3, 1/3 and Pi of rock — so Pi(R) — is a 1/3. So without belaboring it, that's all I'm doing here, is developing some notation. Let's immediately encounter two things you might have questions about. So the first is, that in principle Pi(si) could be zero. Just because I'm playing a mixed strategy, it doesn't mean I have to involve all of my strategies. I could be playing a mixed strategy on two of my strategies and leave the other one with zero probability. So, for example, again in rock, paper, scissors, we could think of the strategy 1/2, 1/2, 0. In this strategy I assign — I play rock half the time, I play paper half the time, but I never play scissors.

So everyone understand that? And while we're here let's look at the other extreme. The probability assigned by my mixed strategy to a particular si could be one. It could be that I assign all of the probability to a particular strategy. What would we call a mixed strategy that assigns probability 1 to one of the pure strategies? What's a good name for that? That's a "pure strategy." So notice that we can think of pure strategies as the special case of a mixed strategy that assigns all the weight to a particular pure strategy. So, for example, if Pi(R) was 1, that's equivalent to saying that I'm playing the pure strategy rock, i.e. a pure strategy.

So there's nothing here. I'm just being a little bit nerdy about developing notation and making sure that everything is in place, and just to point out again, one consequence of this is we've now got our pure strategies embedded in our mixed strategies. When I've got a mixed strategy I really am including in those all of the pure strategies. So let's proceed. I'm going to push that up a little high, sorry. So now I want to think about what are the payoffs that I get from mixed strategies, and again, I'm going to go a little slowly because it's a little tricky at first and we'll get used to this, don't panic, we'll get used to this as we go on and as you see them in homework assignments and in class.

So let's talk about the payoffs from a mixed strategy. In particular, what we're going to worry about are expected payoffs. So the expected payoff of the mixed strategy P, let's be consistent and call it Pi, the mixed strategy Pi is what? It's the weighted average — it's a weighted average or a weighted mixture if you like — of the expected payoffs of each of the pure strategies in the mix. So this is a long way of saying something again which I think is a little bit obvious, but let me just say it again.

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