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This last week on Monday the Israeli Ministry of Communication auctioned 8 bands (each band of 5 mega-herz) for the use of fourth-generation cellular communication. The auction was interesting from a game theoretic perspective, and so I share it with the blog’s readers.

**The rules of the auction are as follows:**

- The auction is conducted on the internet, and each participant sits in his office.
- The minimum bid is 2,000,000 NIS per mega-herz (meaning 10M per band).
- Bids must be multiples of 100,000.
- No two bids can be the same: if a participant makes a bid that was already made by another participant, the e-system notifies the participant that his bid is illegal.

The auction is conducted in round.

In the first round:

- Each bidder makes a bid, which consists of the number of bands it asks for and the price he is willing to pay for each mega-herz. As mentioned above, no two participants can offer the same price.
- The e-system allocates the 8 band between the participants according to the bids: the bidder with the highest bid gets the number of bands it asked for, then the bidder with the second highest bid, and so on, until the 8 bands are exhausted.
- Each participant is told the number of bands it was allocated.
- The price in which the 8th band was allocated is called the threshold and posted publicly.
- The minimum price for the next round is the threshold + 200,000 NIS.

Two governments did not survive this week: the Swedish and the Israeli. Here, in Israel, people are interested in the effect of the coming elections on the financial market. The Marker, the most important national daily economics newspaper, published an article on this issue. The chief economist of the second largest investment house, which handles about 30 billion USD, is quoted as saying (my own translation)

“Past experience shows that most of the time, during six months after elections the stock market was at a higher level than before the elections,” emphasized Zbezinsky (the chief economist, ES). The Meitav-Dash investment house checked the performance of the TA-25 Index (the index of the largest 25 companies in the Israeli stock exchange, ES) in the last six elections. They compared the index starting from 6 months before elections up to six months after elections, and the result was that the average return is positive and equals 6%.

To support this claim, a nice graph is added:

Even without understanding Hebrew, you can see the number 25 at the title, which refers to the TA-25 index, the six colored lines in the graph, where the x-axis measures the time difference from elections (in months), and the year in which each elections took place. Does this graph support the claim of the chief economist? Is his claim relevant or interesting? Some points that came up to a non-economist like me are:

- Six data points, this is all the guy has. And from this he concludes that “most of the time” the market increased. Well, he is right; the index increased four times and decreased only twice.
- Election is due 17-March-2015, which means three and a half months. In particular, taking as a baseline 6 months before election is useless; this baseline is well into the past.
- Some of the colored lines seem to fluctuate, suggesting that some external events, unrelated to elections, may have had an impact on the stock market, like the Intifada in 2001 or the consequences of the Lebanon war before the 2009 elections. It might be a good idea to check whether some of these events are expected to occur in the coming nine months and a half.
- It will also be nice to compare the performance around elections to the performance in between elections. Maybe 6% is the usual performance of the TA-25, maybe it is usually higher, and maybe it is usually lower.

I am sure that the readers will be able to find additional points that make the chief economist statement irrelevant, while others may find points that support his statement. I shudder to the thought that this guy is in charge of some of my retirement funds.

Abraham Neyman and Sergiu Hart are two of the prominent mathematical game theorists to date. Neyman contributed immensely to the study of the Shapley value, stochastic games, and repeated games and complexity. Hart contributed significantly to the study of correlated equilibrium and adaptive processes leading to it, value theory, and formation of coalitions.

Both Abraham and Sergiu will be 66 next year. To celebrate this rare occasion, the Center for the Study of Rationality at the Hebrew University of Jerusalem organizes two conferences, one in honor of each of them. The conference in honor of Abraham will be held on June 16–19, 2015, and the conference in honor of Sergiu will follow on June 21–24, 2015.

Mark the dates and reserve tickets.

Last week I wrote a post about two issues with Elsevier’s e-system, which is the system that all journals run by Elsevier, including *Games and Economic Behavior* and *Journal of Mathematical Economics*, use for handling submissions: the fact that sometimes reviewers can see the blinded comments that other reviewers wrote to the editor, and the user agreement that allows Elsevier to change its terms without notifying the users.

After I corresponded with the editors of *Games and Economic Behavior* and *Journal of Mathematical Economics* and with the Economics Editor of Elsevier, the reason for the privacy breach became clear: the e-system allows each editor to choose whether the blinded comments of one referee to the author and the blinded comments of one referee to the editor will be seen by other reviewers. For each type of blinded comments the editor can decide whether to show it to all reviewers or not. Each editor makes his or her own choice. I guess that often editors are not aware of this option, and they do not know what was the choice that the previous editor, or the one before him, made.

Apparently, the configuration of *Games and Economic Behavior* was to allow reviewers to see only the blinded comments to the author, while the configuration of *Journal of Mathematical Economics* was to allow reviewers to see both types of blinded comments. Once the source of the problem became clear, Atsushi Kajii, the editor of *Journal of Mathematical Economics* decided to change the configuration, so that the blinded comments of reviewers to the editor will not be seen by other reviewers. I guess that in few days this change will become effective. Elsevier also promised to notify all of its journals, in which the configuration was like that of JME, about this privacy issue, and let the editors decide whether they want to keep this configuration or change it. In case this configuration remains, they will add a warning that warns the referee that the blinded comments can be read by other reviewers.

I am happy that the privacy breach came to a good end, and that in the future the e-system will keep the privacy the referees.

Regarding the second issue, Elsevier is not willing to change its user agreement. Reading the user agreements of other publishers, like Springer and INFORMS, shows that user agreements can be reasonable, and not all publishers keep the right to change the user agreement without notifying the users. The Economics Editor of Elsevier wrote: “This clause is not unreasonable as the user can choose to discontinue the services at any time.” As I already wrote in the previous post, I choose to discontinue the service.

*Games and Economic Behavior*, *Journal of Economic Theory*, *Journal of Mathematical Economics*, and *Economics Letters* are four journals that publish game theoretic papers and are published by Elsevier. They all use Elsevier e-system to handle submissions. I already talked in the past about the difficulty of operating these e-systems. Rakesh discussed the boycott against Elsevier. Recently I had some experience that made me stop using the Elsevier’s system altogether, even though I serve on the editorial board of *Games and Economic Behavior*. I will not use Émile Zola’s everlasting words for such an earthly matter; I will simply tell my experience.

**1) The e-system seems to be sometimes insecure.**

I was surprised when a referee with whom I consulted on the evaluation a paper (for GEB) told me that the system showed to him the private message that the other referee wrote to me, and that the same thing happened to him with JME. To prove his point, he sent to me screenshots with the private letter of the other referee for JME.

**2) The user agreement of Elsevier is a contract that one should never agree to sign.**

I guess no one bothered to read the user agreement of Elsevier. I did. The first paragraph binds us to the agreement:

This Registered User Agreement (“Agreement”) sets forth the terms and conditions governing the use of the Elsevier websites, online services and interactive applications (each, a “Service”) by registered users. By becoming a registered user, completing the online registration process and checking the box “I have read and understand the Registered User Agreement and agree to be bound by all of its terms” on the registration page, and using the Service, you agree to be bound by all of the terms and conditions of this Agreement.

The fourth paragraph, titled “changes” says that any change made to the contract is effective immediately, and so it binds you. If you want to make sure they did not add some paragraph to which you disagree, you must read the whole user agreement every time you use the system.

Elsevier reserves the right to update, revise, supplement and otherwise modify this Agreement from time to time. Any such changes will be effective immediately and incorporated into this Agreement. Registered users are encouraged to review the most current version of the Agreement on a periodic basis for changes. Your continued use of a Service following the posting of any changes constitutes your acceptance of those changes.

I contacted Elsevier about the user agreement and got the following response:

The Elsevier website terms and conditions (see http://www.elsevier.com/legal/elsevier-website-terms-and-conditions) cannot be customized upon request; however, these terms and conditions do not often change and notification would be provided via the “Last revised” date at the bottom of this page. The current terms and conditions were Last revised: 26 August 2010.

Well, it is comforting that they did not make any change in the past four years, but will Elsevier’s CEO agree to open an account in a bank that has the “change” paragraph in the contract?

I stopped using the e-system of Elsevier, both as a referee and as an editor.

Roscoff is a village at the north-west corner of France, located on a small piece of land that protrudes into the English canal. Right here, in 1548, the six-year-old Mary, Queen of Scots, having been betrothed to the Dauphin François, disembarks.

As far as I understood, the most common sights in the area are tourists and sea food. As far as I can tell, the main advantage of Roscoff is the Laboratoire Biologique, which is used to host conferences. Every now and then the French game theory group makes use of this facility and organizes a conference in this secluded place. The first week of July was one of these nows and thens. This is my third time to attend the Roscoff conference, and I enjoyed meeting colleagues, the talks, and the vegetarian food that all non-sea-food eaters got.

Here I will tell you about one of the talks by Roberto Cominetti.

Brouwer’s fixed point theorem states that every continuous function $f$ that is defined on a compact and convex subset $X$ of a Euclidean space has a fixed point. When the function $f$ is a contraction, that is, when there is $ρ ∈ [0,1)$ such that $d(f(x),f(y)) ≤ ρ d(x,y)$ for every $x,y \in X$, then Banach’s fixed point theorem tell us that there is a unique fixed point $x*$ and there is an algorithm to approximate it: choose an arbitrary point $x_0 ∈ X$ and define inductively $x_{k+1} = f(x_k)$. The sequence $(x_k)$ converges to $x*$ at an exponential rate.

When the function $f$ is non-expansive, that is, $d(f(x),f(y)) \leq d(x,y)$ for every $x,y \in X$, there may be more than a single fixed point (e.g., $f$ is the identity) and the sequence defined above need not converge to a fixed point (e.g., a rotation in the unit circle).

In his talk, Roberto talked about a procedure that does converge to a fixed point when $f$ is non-expansive. Let $(α_k)$ be a sequence of numbers in $(0,1)$. Choose $x_0 ∈ X$ in an arbitrary way and define inductively $x_{k+1} = α_{k+1} f(x_k) + (1-α_{k+1}) x_k$. Surprisingly enough, under this definition the distance $d(x_k,f(x_k))$ is bounded by

d(x_k,f(x_k)) ≤ C diameter(X) / \sqrt( α_1 (1-α_1) + α_2 (1-α_2) + … + α_n (1-α_n) ),

where C = 1/\sqrt(π).

In particular, if the denominator goes to infinity, which happens, for example, if the sequence $(α_k)$ is constant, then the sequence $(x_k)$ converges to a fixed point. Since the function that assigns to each two-player zero-sum strategic-form game its value is non-expansive, this result can become handy in various situations.

This is a good opportunity to thank the organizers of the conference, mainly Marc Quincampoix and Catherine Rainer, who made a great job in organizing the week.

Yesterday the Israeli Parliament elected the state’s new president. This position is mostly ceremonial, and does not carry any significant duty. Nevertheless, quite a few people, mostly current and past Parliament members, wanted to get this job. The race was fruitful, as a couple of them were forced to withdraw their candidacy after secrets of weird sex stories and bribes were published in the press.

Three main candidates survived until the final stage, Rivlin, Itzik, and Shitrit (in addition to two candidates who did not stand much chance). Election is done by a two-round system: Each of the 120 Parliament members secretly votes for a candidate. If no candidate gets more than 60 votes, then all candidates except the leading two leave the arena, and the Parliament members secretly vote to either one of the two leaders. The candidate who got more than 60 votes is the new proud president of the state.

Politics was ugly. Rivlin, who is a member of the largest party, was the leading candidate, but the Prime Minister, who comes from the very same party, was against him. Many members of the opposition voted for Rivlin. In the first round, Rivlin and Shitrit got the highest number of votes, but none of them had a majority of votes. In the second round, Rivlin won.

What I found interesting in this process is what one of the Parliament members of the second largest party said. He said that in the first round some members of his party voted for Shitrit, but then, in the second round, they voted for Rivlin. Why? “It was a tactical vote.” This way they ensured that the final election will be between Rivlin and Shitrit and *not *between Rivlin and Itzik. Politics is ugly, but at least as long as the election process does not satisfy Independence of Irrelevant Alternatives, we do not have dictatorship.

Abraham Neyman had numerous contributions to game theory. He extended the analysis of the Shapley value in coalitional games with player set which is a measurable space, he proved the existence of the uniform value in stochastic games, and he developed the study of repeated games with boundedly rational players, among others. Abraham Neyman was also one of the founding fathers of the Center for Game Theory at the State University of New York at Stony Brook, which hosts the annual conference of the community for the past 25 years.

The *International Journal of Game Theory* will honor Abraham Neyman on his 66th birthday, which will take place in 2015, by a special issue, see the announcement here. Everyone is encouraged to submit a paper.

This post is dedicated to a new and important result in game theory – the refutation of Mertens’ conjecture by Bruno Ziliotto. **Stochastic games** were defined by Shapley (1953). Such a game is given by

- a set Z of
**states**, - a set N = {1,2,…,n} of
**players**, - for every state z and every player i a set A_i(z) of
**actions**available to player i at state z. Denote by Λ = { (z,(a_i)_{i ∈ N}) : a_i ∈ A_i(z) for every i} the set of all pairs (state, action profile at that state). - for every player i, a
**stage payoff function**u_i : Λ → R, and - a
**transition function**q : Λ → Δ(Z), where Δ(Z) is the set of probability distributions over Z.

The game starts at an initial state z^1 ∈ Z and is played as follows. At every stage t, each player i chooses an action a_i^t ∈ A_i(z^t), receives a stage payoff u_i(a_1^t,…,a_n^t), and the play moves to a new state, z^{t+1}, that is chosen according to q(z^t;a_1^t,…,a_n^t).

In this post I assume that all sets are finite. The N-stage game is a finite game, and therefore by backwards induction it has an equilibrium. As I mentioned in my previous post, the discounted game has an equilibrium (even a stationary equilibrium) because of continuity-compactness arguments.

When studying dynamic interactions, economists like discounted games. Existence of equilibrium is assured because the payoff is a continuous function of the strategies of the players, and construction of equilibrium strategies often require ingenious tricks, and so are fun to think about and fun to read. Unfortunately in practice the discounted evaluation is often not relevant. In many cases players, like countries, firms, or even humans, do not know their discount factor. Since the discounted equilibrium strategies highly depend on the discount factor, this is a problem. In other cases, the discount factor changes over time in an unknown way. This happens, for example, when the discount factor is derived from the interest rate or the players monetary or familial situation. Are predictions and insights that we get from a model with a fixed and known discount still hold in models with changing and unknown discount factor?

To handle such situations, the concept of a **uniform equilibrium** was defined. A strategy vector is a **uniform ε-equilibrium **if it is ε-equilibrium in all discounted games, for all discount factors sufficiently close to 1 (that is, whenever the players are sufficiently patient). Thus, if a uniform ε-equilibrium exists, then the players can play an approximate equilibrium as soon as they are sufficiently patient. In our modern world, in which one can make zillions of actions in each second, the discount factor is sufficiently close to 1 for all practical purposes. A payoff vector x is a **uniform equilibrium payoff** if for every ε>0 there is a uniform ε-equilibrium that yields payoff that is ε-close to x.

In repeated games, the folk theorem holds for the concept of uniform equilibrium (or for the concept of uniform subgame perfect equilibrium). Indeed, given a feasible and strictly individually rational payoff x, take a sequence of actions such that the average payoff along the sequence is close to x. Let the players play repeatedly this sequence of actions while monitoring the others, and have each deviation punished by the minmax value. When the discount factor is close to 1, the discounted payoff of the sequence of actions is close to the average payoff, and therefore the discounted payoff that this strategy vector yields is close to x. If one insists on subgame perfection, then punishment is achieved by a short period of minmaxing followed by the implementation of an equilibrium payoff that yields the deviator a low payoff.

For two-player zero-sum stochastic games, Mertens and Neyman (1981) proved that the uniform value exists. Vieille (2000) showed that in two-player non-zero-sum stochastic games uniform ε-equilibria exist. Whether or not this result extends to any number of players is still an open problem.

Why do I tell you all that? This is a preparation for the next post, that will present a new and striking result by a young French Ph.D. student.

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