There's a flip side to that. There's a what's called a

dominated strategy. A dominated strategy is a strategy that

never does better than another one. So we compare two different strategies,

and if one always does worse than the other, this strategy is dominated.

So of course a rational player would never opt for that strategy.

And again, it's easy for yourself. If you're trying to maximize your own

profit, why would you choose a dominated strategy?

But what's more interesting, and what takes more thinking, is

that a rational rival will never play a dominated strategy, right?

So even if a dominated strategy by your rival would give

you massive profits you should never count on the other

player choosing a dominated strategy if he's interested in maximizing profits,

okay? So if we want to solve the pizza war, and

basically try to predict what's going to come out, what the prices are going

to be, we can try to to eliminate the dominated

strategies in this case, okay? So if we just have a look we can

eliminate the dominated strategies one by one.

Okay, let's go back and see how we can do that.

So if Piccola Osteria would charge a high price, okay we're here, Pizza Rosso can

choose high price and get 6, a medium price and get 7

and a low price and get 3 and a half. So the best strategy here would be a

medium price. If Piccola Osteria chooses a medium price

then the best response, the best strategy that Pizza Rosso can choose

is to choose a medium price again. If the Piccola Osteria chooses a low

price, the best strategy by Pizza Rosso is actually to choose a high price,

right, 3.6 versus 3 versus 2 and a half. Why do they do it?

They've basically given up the market for the price sensitive consumers and they

only sell to their loyal consumers at a high margin.

Now, if you look at this, what's interesting is that if you compare a low

price, and a medium price. It's always better for Pizza Rosso to

choose a medium price rather than a low price.

And what that means is that there is no circumstance, no strategy by

Piccola Osteria, under which Pizza Rosso would want to choose a low price.

Now what we can do with this strategy is basically just eliminate it.

It's not going to happen, right? There's never going to be a low price set

by Pizza Rosso. We can do it exactly the other way

around, and given this is a symmetric problem, we'll find that also for Piccola

Osteria charging a low price is never going to be

a viable strategy, okay? So, we eliminate these dominated

strategies. We can eliminate them one by one, and we

will continue with that until we can find a solution to the game or you just can't

eliminate more. And let's go back to the game

here. Once we've simplified this game, let's

have a look again. If Piccola Osteria charges a high price

we've already figured out that it's better for Pizza Rosso to charge a medium

price than a high price. Because 6 is less than 7.

If Piccola Osteria charges a medium price, then it's better for Pizza Rosso

to charge a medium price than high. Which means that here charging a high

price is dominated again. And by the same token, we can do the same

thing for this simplified game and eliminate the high price for Piccola

Osteria as well. And this is going to let us solve this

game, make a prediction of what the outcome of this game is.

Namely, both Pizza Rosso and Piccola Osteria charging a medium price

and both making profits of 5000 euros. In this video we analyzed very briefly

the strategic problem of choosing the right price

and we structured this problem with the help of a matrix that was slightly more

complicated than the first one we saw. We used the process of iterated

elimination of dominated strategies, so we kept on cutting up the game to make it

simpler and simpler. To eventually find the optimal strategies

for Piccola Osteria and Pizza Rosso respectively.

So in the next video we'll go on and demonstrate to you a more formal and a

more precise way of thinking about optimal strategies.

So please stay on to, to go on to the next video.

But first, let's have a look at the exercise and figure out all the dominated

strategies in these exercises. Again, any case you'll find help in our

forum. Thanks very much and see you in the next video.