Game Theory and the Kyoto Protocol

I thought I'd take a break from depressing news about the financial markets and tackle depressing news about the environment. I just read an article in the Economics Focus section of The Economist: Playing games with the planet, Sep 27, 2007. In that article, The Economist discusses research that analyzes the current impasse over the Kyoto Protocol from the perspective of Game Theory.

Basically, the Kyoto Protocol resembles the Prisoner's Dilemma -- a game where players should cooperate from the perspective of globally optimal payoffs, but the players have incentives that would cause them to defect from cooperation and lead to mutually harmful payoffs. The Prisoner's Dilemma can be ameliorated by converting it from its static form to a repeated game. In the repeated Prisoner's Dilemma, players can see what the other players did during previous rounds and 'punish' them for defection.

As Robert Axelrod (an American political scientist -- who took the science part seriously) and others have demonstrated, the best strategy that emerges for repeated PDs is the so-called 'tit-for-tat' strategy. In most cases, that strategy recommends that (a) players should start by cooperating, (b) if some subset of players defect, then they should be punished by the cooperating players but that this punishment should last no more than the time or magnitude required to get players back in line, and (c) hopefully, all players will return to cooperation.

Michael Liebreich, of New Energy Finance, argues that -- while the current impasse over Kyoto seems to be more like the static PD -- the situation should be seen as being more like the repeated PD. Mr. Liebreich offers hope that, with some alterations to the Kyoto and post-Kyoto attempts to control global warming, something akin to 'tit-for-tat' can bring about a global regulatory mechanism to combat global warming.

Although I admire Mr. Liebreich's no doubt good intentions, I think he is being too optimistic. The flaw with his thinking is that there is another framework to consider beyond Game Theory: externalities. As The Economist cheekily points out:

At any given summit on climate change, it is never long before some politician declares how “urgent” or “vital” or “imperative” it is to stop the planet from overheating. And yet few governments are willing to tackle the problem by themselves. In practice, what these impassioned speakers usually mean is that it is urgent—no, vital!—no, imperative!—for all countries but their own to get to grips with climate change.

This quote nicely sums up the problem with economic externalities (aka, external costs and benefits): The benefits of controlling global warming would be shared with countries not cooperating, but the costs will be borne only by those who cooperate. In other words, countries like the U.S. that refuse to cooperate with global warming treaties will free-ride off the countries that are doing something about global warming.

The logic of repeated PD and 'tit-for-tat' rests on the fact that, in those idealized scenarios, players can monitor defection and can adequately punish defectors. But with externalities, it would be difficult to monitor who benefits even though they are free-riding, and it would be even more difficult to punish free-riders without causing serious lasting damage to both the cooperators and defectors.

For example, let's say Denmark decides to punish America for not cooperating on Kyoto or Kyoto-like global warming control schemes. How would it do that? Dump toxic waste to get back at America? But that would cause lasting damage to not only America and Denmark but to the global environment itself ... precisely what they were trying to avoid!

Unlike 'tit-for-tat' of laboratory repeated Prisoner's Dilemma, in the real world global warming game, it would be hard to limit the punishment so that it would send a strong enough signal to cooperate but not so strong as to be destructive to everyone involved.

Back to the drawing board, I guess, for utilizing Game Theory for negotiating global warming treaties.

eBay 'Sniping' and the Power Law

'Sniping' -- where a bidder waits until the last moments of an auction to place a bid -- is a tactic used by some eBay auction aficionados to gain an upper hand in bidding for an item.

According to an article (June 23, 2006) in the New Scientist, two South Korean physicists, Byungnam Kahng and Inchang Yang, found that a power law equation explained the observed patterns in eBay auction bidding. The power law reflects the fact that most of the bidding for items tend to take place near the end of an auction. Thus, according to their article in Physical Review E (vol. 73, p 067101, 2006), 'sniping' is a rational strategy to employ in eBay bidding.

This research builds on the work of experimental economist / game theorist, Alvin Roth. His article in the American Economic Review (vol 92, p 1093, 2002) came to similar conclusions.

Traveler's Dilemma: 'Irrational' Game Theory

I just read an interesting article in Scientific American on game theory -- The Traveler's Dilemma (June 2007) -- written by Kaushik Basu, the economist who created the aforementioned game paradigm. The Traveler's Dilemma, as the Sci Am article describes it, is very similar to the Prisoner's Dilemma except that the payoff matrix of the typical PD scenario is a kind of subset of the payoff matrx of TD (specifically, when the players are restricted to the lowest two strategic choices, TD is equivalent to PD).

The Traveler's Dilemma (TD) basically involves the following scenario: Two players can individually choose a figure (usually couched in terms of some monetary amount or price) without in any way conferring or coordinating with each other. If both players choose the same amount, then they will each be paid that amount (hence, like Prisoner's Dilemma (PD), there are meta-incentives to 'cooperate'). But if they choose different amounts, then the player who chooses the lower amount will get that amount plus some reward/bonus while the player who chooses the higher amount will get the lower amount minus a penalty (which is usually the same size as the reward). Thus, as in the better known PD, the TD's payoff structure provides myopic incentives for players to 'defect' and try to undercut one another.

Under the standard assumptions of 'rationality' (I put it in quotes here because I find economists' and game theorists' definition of rationality to be too straight-jacketed and narrow) and the logic of backward induction (a solution concept in game theory), the players in TD should defect and choose the lowest number possible (similar to the result in PD). Of course, this result is to the deteriment of both players since both of them would have gotten a higher payoff if they had 'cooperated' and chosen a high amount.

Prof. Basu argues -- providing ample experimental/empirical evidence and common sense observations -- that, in reality, the theoretical result of TD does not hold. In practice, most players who participate in TD inspired game theoretical experiments tend to choose figures in the high range of options. These empirical results fly in the face of the standard assumptions of 'rationality' (although, as Prof. Basu helpfully points out, it is consistent with a sort of meta-rationality) in game theory and economics.

The article makes some other interesting observations. One of them are the results of an experiment carried out by economist/game theorist, Ariel Rubinstein. Rubinstein found that people who based their decisions in Traveler's Dilemma style experimental games using strategic reasoning or more formal 'rationality' took the longest time to respond to prompts. Conversely, those who made decisions on spontaneous 'emotional' or untrained intuitive responses or who made "random" (i.e., inexplicable or perhaps even crazy) choices tended to take the least amount of time in making decisions.

Another interesting observation made in the Scientific American article is whether or not the economist's/game theorist's concept of rationality needs to be modified and expanded. Prof. Basu argues that the standard notions of rationality are inadequate to explain the empirical tests of the Traveler's Dilemma as well as the Prisoner's Dilemma (especially, iterated PD). The problem that the article zeros in on is the assumption that rationality is "common knowledge" in its philsophical, formal logic sense of that term (i.e., that each of the players in TD know that the other player will act 'rationally' and they are each know that the other knows). Prof. Basu argues that people may not always conform to this standard assumption, especially in repeated games such as iterated (or repeated) Prisoner's Dilemma.

As the article points out, perhaps these observations from empirical tests of TD and PD will lead to a rethink of game theoretic logic. Perhaps there is a "meta-rational" consideration(s) that lead people to (without direct collusion and communcation) choose mutually beneficial options. This type of reasoning should give us some reasons to hope in humanity at large.

Mimicking Soros, Dangers of Low Volatility, & Collusion in the Music Industry

I've noticed several interesting articles in The Economist magazine recently that are worth noting.

In his/her column, Soros on the cheap (April 4, 2007), Buttonwood makes the case for currency trading as a valuable addition to an investment portfolio. In particular, currency trading based on exchange rate models that take into account the 'carry trade' (an arbitrage technique which is similar to short selling), momentum, and purchasing power parity, have shown themselves to be profitable.

In last week's column, Sting in the tail: Is low volatility making the world too complacent about risk? (April 12, 2007), Buttonwood makes an even more convincing argument that instruments, practices, and institutions in the financial markets that leads to a relatively low volatility environment most of the time may wind up increasing 'tail risk' (i.e., extreme risk). Low volatility may be a 'false dawn' -- or perhaps even a cruel joke being played by the forces of market randomness -- that lull investors and traders into a false sense of security ... suckering people along into making bad financial decisions until catastrophe strikes.

Finally, in the March 29, 2007, 'Economics Focus' column, Silent orchestration: Can record companies act in concert, even without agreeing to do so?, The Economist examines the possibility that there is tacit collusion in the music industry by applying the logic of game theory. This type of analysis is very important to monopolies, antitrust, and competition laws & regulations. As the column points out, it is difficult to hold a cartel together (tacit or explicit), but, using game theoretic methods, it isn't impossible.

Book Review: The Mathematics of Poker

The game of poker is a fascinating mix of mathematics and psychology. The combinatorial and probabilistic nature of the shuffling and dealing of cards -- as well as the uncertainty associated with competing against people with differing personalities and backgrounds -- suggests that a good poker players should be, at least on an intuitive level, good mathematicians. The need to make major decisions under pressure also suggests that poker is a case study in applied psychology.

As someone fascinated by both econophysics and poker, I drew the conclusion that mathematics could be fruitfully applied to both the more obviously 'mathematical' aspects of poker as well as the psychological/decision-making aspects of poker (via game theory). With the ongoing popularity of poker on television, on the internet, and in card rooms, as well as a growing body of poker 'literature,' I naturally expected there would be at least a few books available that took on poker mathematics. Sadly, the vast majority of poker books only deal with 'math' on a purely computational or numerical basis -- i.e., they explain or offer up various types of odds associated with poker and not much beyond that -- rather than applying the analytical tools provided by mathematics in its more purer sense.

The Mathematics of Poker by Bill Chen -- a mathematician, part time pro poker player, and full time financial quant at Susquehanna International Group -- and poker pro, Jerrod Ankenman, finally addressed many of my musings on the intricate relationship between poker and mathematics. Although this book is not for everyone, especially those who are faint of heart when it comes to equations and formulae, it should be highly useful to quantitatively minded poker players and fans as well as to finance types that may (or may not) be surprised to find so many commonalities between poker and quantitative finance.

One of the interesting attributes of this book is that, despite a plethora of equations and semi-formal mathematical expressions that very few poker players will actually work through at the table, the authors take the position that the mathematical analysis and reasoning they use in this book is designed to make the reader more profitable poker players. This practical stance actually enhances the intellectual credibility of this book: While the more self-consciously 'intellectual' works that tackle 'poker' deal merely in abstract scenarios that differ dramatically from real-world poker, Chen & Ankenman's book offers up analysis (even when using 'toy games') that is tied to what poker actually looks like.

Some of the high points of this book are: the book's explanation of how mathematical statistics and probability -- especially Bayesian approaches -- might apply to poker decisionmaking (Parts I & IV), the application of mathematical game theory to the game of poker (Parts II, III, V), the concept of "effective tournament size" (where tournament payout structure alters the number of 'double ups' needed to make money in a poker tournament) (Part V), a quantitative approach to poker backing agreements (Part IV), the important role that exponentials and logarithms play in the mathematical analysis of poker (Parts III and IV), risk management for poker players (Part IV), and the scientific approach to the 'art' of hand reading (i.e., making an educated guess of the distribution of cards your opponent may have) (Part II).

An especially fascinating aspect of this book is how the authors make some interesting analogies that connect poker to quantitative finance. The idea of maximizing logarithmic utility, which is at the heart of a lot of quantitative finance, is discussed in detail in connection with profitability in poker play (I should note that the authors call this concept the 'Kelly Criterion' or 'Kelly betting,' but for reasons beyond the scope of this blog post, I don't quite agree with this characterization because the Kelly Criterion is a much deeper concept, IMHO, than maximizing log utility). The book also bring in other concepts from quantitative finance and financial economics into poker analysis, including the Sharpe ratio, financial options (real options), and modern portfolio theory.
The best part of this book was its explanation of the 'risk of ruin' and how it relates to poker play over time. I have read many books on probability, statistics, general mathematics, and gambling over the years and I have always felt frustrated by the lack of a clear explanation of the rather useful but basic concept of the risk of ruin. The Mathematics of Poker gives, by far, the best explanation of the risk of ruin I have ever read.

Although this book makes many very excellent points and should be a valuable addition to anyone intersted in the subject matter, this book does have some flaws. One of the most obvious flaws is that it has a number of typos and errors. I wished the authors or the publishers had invested in LaTEX typsetting (it doesn't appear that way to me). Having pointed this out, however, I should, in the book's defense, also note that: (a) most of the errors are minor and of a typographical nature rather than a substantive nature, and (b) the authors and publishers are putting out an errata and have been making corrections to new printings of the book (details on this can be found on the book's website) -- which are the responsible things to do (that many others neglect to do in the technical publishing world). Another criticism along the same lines is that some of the notation is confusing (e.g., the Greek symbol for 'alpha' means radically different things in different parts of the book) and should have been better thought out.

The only other major criticism I have the book is 'Part III: Optimal Play.' Although game theory is utilized throughout the book, Part III is where the bulk of the application of game theory to poker takes place. So it frustrated me to find this part to be tedious even to someone who is an avid reader of highly mathematical and technical material. Having said that, however, I should note that there are definitely interesting and worthwhile points of wisdom made in Part III. Furthermore, the other parts of the book are so interesting in and of themselves that Part III can be safely skimmed in order to 'enjoy' (to the extent one can 'enjoy' a mathematics book) the book as a whole.

The final point that needs to be made about this book are the mathematical prerequisites needed to read this book. The authors state that they have kept the prerequisites to a minimum and that someone with a very solid high school college prep mathematics knowledge base can understand this book. Although I think the authors are sincere in their claims, I think this book would be a challenging read for those who are limited to that criteria. I think a more realistic mathematical prerequisite for reading this book is either someone who has had at least some education in calculus (which the authors occassionally throw in) or someone who regularly reads math books -- they could be 'pop' math books -- that have equations and algebraic manipulations in them. If you have that level of mathematical sophistication -- which is still a fairly low standard -- you should do alright with going through the type of reasoning used in this book.

In summary, I believe that this book does an excellent job of finally addressing what had been a glaring omission in the poker literature: the application of mathematics (as opposed to just numbers and computations) to poker. As Chris Ferguson, a World Series of Poker main event champion and a holder of a PhD in computer science from UCLA, said in his endorsement of the book, "If I ever find myself teaching a poker class for the mathematics department at UCLA, this will be the only book on the syllabus."

Bill Chen & Jerrod Ankenman's book may do more than offer up a route to intellectual exploration, however. As Jeffrey Yass, Bill Chen's boss at Susquehanna, states "In the same way that quants and mathematicians took over Wall Street in the late 80's, mathematical methods will dominate poker in years to come."