What you should do after today's lecture (October 5)
Starting from the last thing we did: express your preferences between those two pairs of lotteries. Do it seriously, and don't forget that millions of Euros are life-changing amounts of money.
Of course you should read basic definitions, starting from section 2.2, and the theorem from the book (the teorem is on page 62). Skip section 2.3 (in general you will want to skip the sections we didn't mention at all in class). Sections 2.4 to 2.6 contain the core material. You should try the exercises marked with "a". I will do 2.6.2 and 2.6.3 next time. And we'll prove the theorem then. As I said, try to show that EU implies the three axioms.
Important: read the "independence condition" on page 64 and show (it is not so difficult) that EU implies it. This is immediate after exercise 2.6.6, but since you may find that hard and it is fundamental,
for its (very) detailed solution.
For your convenience the Allais experiment we discussed in class is
We will not finish the proof of the Eu Theorem (extension from [m,M] to the whole line) in class since some of you would get bored. Whoever is reading the proof from the book is welcome to come see me for questions and clarifications.
Call as usual x our benchmark two-outcome lottery: alpha with probability p and beta with probability 1-p; so Ex=p*alpha+(1-p)*beta. Start with the picture
which displays a concave utility function U and draw EU(x), U(Ex), CE(x) and pi(x) (the risk premium).
Also as usual do the exercises marked "a". In particular do example 3.2.2, exercise 3.3.4.
You can also do exercise 2.5.6 if you can compute derivatives (which you should).
And obviously there is the coach problem: that one you can discuss collectively.
A summary of what we did about risk aversion is
Please revise logs and exponents (basic equations and derivatives)
Problem set solution
Today's lecture was based on Osb-Rubi section 3.1. We have done def. 32.1 and you can read the obvious def 32.3 - and also the paragraph which follows it.
Proposition 33.1 is about existence, just read the assertion.
Formula (33.2) is crucial so I have written more detailed notes leading to it in
this file. Lemma 33.2 also is central for finding equilibria, and its derivation is also included in that file.
You can read example 34.1, battle of sexes, which we have done already.
If you feel you need more you can go through sections 4.1-4.5 of Osborne. In any case do sections 4.6 and 4.8 there. Section 4.10 also is good as practice.
On pure strategy equilibrium Osborne chapter 3 has nice illustrations. We will do Cournot (section 3.1) in class.
You can study auctions from section 3.5 if you cannot do it from Osb-Rubi page 18. We will also cover section 3.6 in class.
Analysis of Rock-paper-scissors is in a file in the course page. In that page you can also look at
The bus game, A two-by-two simple game (hiring, from Gibbons), A couple of exercises on normal form games by Joel Watson and The penalty game.
Do as much as you can, we'll work together in class next Tuesday. Have a good week end, SM.
Solution to Accident Law example here.
For the next valuation (due next Tuesday) I would like you to find the equilibria in the two games contained in
here. If you have time please also do exercise 97.1 from Osborne book. You can work on your own or in a small group, as you prefer.
We have done the mixed extension of general games as in
On zero-sum games we have covered the material contained in these files:
Main result on zero-sum equilibria and
Mixed extension of zero-sum games.
On Friday we will complete our glance at correlated equilibria.
Don't forget to look at the definition of conditional probability - either in your statistics textbook or on Wikipedia!
First of all: the lecture of next Friday is re-scheduled for Monday December 3 at 12am, Aula 3.
We have seen that in games it may be better not to have more information. This cannot happen in one-person decision problems, as you can see in
For next Tuesday I would like you to study the game in
Of course you should look at the correlated equilibrium theory in section 3.3 of Osborne-Rubinstein. The basic definitions are in Osborne-Rubinstein section 3.3.
This file contains some clarifications you may find useful.
The example we have worked out in today's lecture is contained
The definition of Bayes-Nash equilibrium can be taken from Osborne. Another possible source is a chapter of a book by Tadelis which I temporarily have uploaded in download directory of my page (email me if you have trouble reaching it). As more examples, besides those we have done in class today, you can do, from Tadelis, "Study groups" sec. 12.2.2 and "Committee voting" sec. 12.4. As an (easy) exercise you can solve the game in
To the exercise above please add the one contained in
You find the duels game in the course page (course material). There you also find an example of a game with simultaneous moves, "bank deposit game", which you may want to look at as well.
Other interesting examples are in Osborne ch.6: for Oligopoly the equilibrium is computed in sec.s 6.2.1-6.2.2 (the rest is analysis of equilibrium which is more economics than games, it depends on your interests...). Section 6.3 models a situation where two interest groups offer money to members of parliament to influence outcome of a vote to their favor.
Please let me know if we meet next Tuesday or Friday!
First let me give you the solutions of the homework you handed today. The Bertrand game solution is
that of the public good provision is
and the solution to the duels game is
On the examples and exercises of the few pages of Osborne-Rubinstein we covered today I have written a few notes to ease your reading. The file is
Please finish reading the Nucleolus derivation of the Airport Game, and also look at the Core of that game (end of file). As I said in class our last two lectures will take place next Thursday at 12 pm (Aula II) and Friday at 8am (Aula Colletti). See you on Thursday.
Here is a reader's guide to what we have done today. Definition of repeated game from OR (137.1). You may want to read sections 8.1 and 8.2 there as well. Evaluations of streams are in section 8.4, and preliminarily you may look at ponits 1 and 2 of my file "Preliminaries on discountijng flows". We did secion 8.4. And we did Proposition 157.2 (you may want to read the example following it as well. For the Prisoner's Dilemma we did essentially (more than) Osborne 14.7.1 and 14.10.1. The one-deviation property is in section 14.9 but it suffices to know that it holds, the argument is based on the result in my file that far distant future "does not count" under the discounted sum crtiterion. Summping up we have seen that in the PD - as in any other game with unique NE - the only equilibrium in the T-repeated game consists of repetitions of that NE; and that in the infinitely repeated version of the game cooperation can be sustained if the players are patient enough. Tomorrow we will look at what payoff profiles can be attained in repeated games more generally.
As an exercise you can answer the easy questions in the file "Long Run Cooperation in oligopoly" in the course page.
The final problem set, due January 28, consists of: the oligopoly exercise mentioned above; the cooperative game exercise contained in
this file; and the exercise contained
here where I ask you to draw the normal form of an extensive form game and compute its mixed equilibrium.
Lastly, for the more curious, the assignment is the following: study Proposition 156.1 in Osborne-Rubinstein (which we have not quite completed in class); and make sense of the inequality contained in the proof in the third line from the end (contained in the sentence starting with "To deter such a deviation...").
Bye bye, see you then. I am leaving on Sunday, I hope I will enjoy my break.