Gladiators, Pirates and Games of Trust
Page 14
Who is the lesser player? It’s the player whose chances of winning a bet are smaller than his opponent’s (even if only by a fraction), or the player who has less money (and fewer opportunities to correct losses) than the other player.
When you play against the house, your place on that scale is quite clear. The house always has the edge (that’s what the single zero and the double zero on the roulette wheel are for), the experience, and the money you don’t have.
Let me warn you again, though. Do not gamble! This is perhaps the best mathematical advice I can give you (except if you’re doing it for fun and don’t mind paying for it by losing – and, if that’s the case, I’d suggest you decide on the price you’re willing to pay before you play, and stick to it). Perhaps you’ll be surprised to know that I can intuitively explain why the bold play is the optimum strategy that would maximize your chances of making that $10.
To simplify the explanation, let me present another problem that will shed a clear, bright light on the roulette table question.
Imagine that I happen to come across the basketball genius Michael Jordan and he agrees to shoot some hoops with me. At the moment in question, neither of us is an active NBA player and we both have plenty of free time. Certain of his skills, MJ generously lets me call the score we aim for. What would you suggest? I hope that the answer is clear. The best thing for me to do is to call the whole thing off, give Michael a hug and call it a tie (though it would be very foolish to pass up the chance to play against my hero). The second-best solution is to play for a single point. I mean, miracles do happen. I could shoot, and the ball could be kind to me and swoosh right in, while MJ might miss his shot (it happens to the best of us).
If I choose to play for two or three points, my chances of winning would drop depressingly low; and if we play on, I’ll most certainly lose. The ‘law of large numbers’ predicts that, in the long run, what’s expected will happen. If we play for one point, I can at least fantasize about beating the Michael Jordan in basketball. Dreaming is free.
If we return to the casino question, let me remind you that the roulette contains zeroes, and they tip the scales in the house’s favour and make the whole game unfair (to me). On the qualitative level, betting against the house is the same as playing basketball against Michael Jordan. The house is the better player, so it would be advisable for me to play as few times as possible, because in the long run the house always wins.
Casino experts or reasonable mathematicians may wonder what happens if, having $4, we bet a single dollar first and then play the following strategy: if we win and have $5, we bet on $5, and if we lose and are down to $3, we switch to the aforementioned ‘bold game’ strategy. The answer is that this strategy gives the same winning probability as playing the bold game right from the start. In any event, this remark is for experts only.
On the other hand, if your goal is to spend some quality time in the fancy casino, the bold game isn’t your best option, because it might motivate the house detective to show you to the door after just one game. If spending time in the casino is your ultimate goal, I’d suggest playing cautiously – bet a single dollar each time and take long breaks. This isn’t the brightest strategy, but it’s highly effective as a way to spin out your time and money.
Let me sum up this chapter with an insight attributed to the British statesman David Lloyd George: ‘There is nothing more dangerous than to leap a chasm in two jumps.’
Conclusion
GAME THEORY GUIDELINES
Game Theory deals with formalizing the reciprocity between rational players, assuming that each player’s goal is to maximize his or her benefit – in terms of such benefits as money, fame, clients, more ‘likes’ on Facebook, pride and so on. Players may be friends, foes, political parties, states or any other kinds of entity with which you can be interactive.
When you’re about to make a decision, you should assume that, in most cases, the other players are as smart and as egotistical as you are.
When entering negotiations, you must take three key points into consideration. You must be prepared to take into account the possibility of ending the talks without an agreement; you must realize that the game may be repeated; and you must deeply believe in your own stands and stick by them.
Playing rationally against an irrational opponent is often irrational, and playing irrationally against an irrational opponent is often rational.
Try as much as you can to guess what your opponent will do by trying to walk in their shoes. You are not him or her, however, and you can never know exactly what makes them tick: you’ll never have a complete handle on what they’ll do and why.
Remember that to explain is much easier than to predict. Most things are more complicated than you think, even if you think you understand this sentence.
Always take into account the human unwillingness to accept injustice, as well as the significance of honour.
Beware! The mathematical solution of a game often ignores such important things as envy (every time a friend succeeds, I die a little), insult, schadenfreude, self-respect and moral indignation.
Motivation may improve strategic skills.
Before making any decision, ask yourself what would happen if everyone shared your views … and remember that not everyone does share your views.
Sometimes ‘ignorance is bliss’: it may happen that the least-knowledgeable player makes the highest profit when competing against extremely clever, all-knowing players.
When each player plays his or her own best choice and takes no care at all of the consequences of their action on other players, this may result in a catastrophe for all. In many situations egoistical behaviour is not only morally problematic but also strategically unwise.
Contrary to the popular belief that having more options is a better option, it may happen that narrowing down the number of choices will improve the result.
People tend to cooperate when faced with the ‘shadow of the future’ – when further encounters are expected, we change the way we think. When the game is played over and over again, stick to the following: ‘Play nice. Never be the first to betray, but always react to betrayals. Avoid the pitfall of blind optimism. Be forgiving. Once your opponent stops betraying, stop betraying too.’
Bear in mind the words of Abba Eban: ‘History teaches us that men and nations behave wisely once they have exhausted all other alternatives.’
Study the possible permutations of success and failure that result from particular moves in the game in question. Learn the consequences of both honesty and duplicity, and the risks involved in trust.
Don’t get sidetracked by the fascination of complexity if your simple aim is to win. As Winston Churchill said, ‘However beautiful the strategy, you should occasionally look at the results.’
Giving up on economic/strategic thinking and simply trusting the other player is time and again the reasonable thing to do.
Moral qualities such as honesty, integrity, trust and caring are essential for a sound economy and a healthy society. There’s a question as to whether world leaders and economic policymakers are endowed with such qualities, which give you no edge whatsoever in political races.
If you are the ‘lesser player’, you should play as few games as possible.
Trying to avoid risk is a very risky course of action.
REFERENCE NOTES
Chapter 3: The Ultimatum Game
Page 15: An extensive review of the Ultimatum Game can be found in Colin F Camerer, Behavioural Game Theory, Princeton University Press, NJ, 2003.
Page 15: The article by Werner Guth, Rolf Shmittberger, and Bernd Schwarze is ‘An Experimental Analysis of Ultimatum Bargaining’, Journal of Economic Behaviour and Organization, 3:4 (December), pp. 367–88.
Page 22: Maurice Schweitzer and Sara Solnik wrote up their study on the impact of beauty on the Ultimatum Game in ‘The Influence of Physical Attractiveness and Gender on Ultimatum Game Decisions’, Organizational Behaviour and H
uman Decision Processes, 79:3, September 1991, pp. 199–215.
Page 23: The Jane Austen quotation is from Pride and Prejudice, volume 1, chapter 6.
Chapter 4: Games People Play
Page 39: Martin Gardner’s thoughts on Game 5 can be found in Martin Gardner, Aha! Gotcha: Paradoxes to Puzzle and Delight, W H Freeman & Co. Ltd, New York, 1982.
Page 40: See Raymond M Smullyan, Satan, Cantor, and Infinity: And Other Mind-boggling Puzzles, Alfred A Knopf, New York, 1992; and Dover Publications, 2009.
Chapter 6: The Godfather and the Prisoner’s Dilemma
Page 88: Robert Axelrod’s The Evolution of Cooperation, Basic Books, 1985; revised edition 2006.
Chapter 7: Penguin Mathematics
Page 99: A note on Strategy 1, War of Attrition: In 2000 I published a paper, with Professor Ilan Eshel, that deals with the mathematical aspects of volunteering and altruism. The mathematics is not easy, but if you’re interested the paper is free on the Internet: just Google ‘On the Volunteer Dilemma I: Continuous-time Decision Selection 1(2000)1– 3, 57–66’.
Page 103: See John Maynard Smith, Evolution and the Theory of Games, Cambridge University Press, Cambridge, 1982.
Chapter 8: Going, Going … Gone!
Page 107: The article referred to here, published in 1971, is Martin Shubik, ‘The Dollar Auction Game:
A Paradox in Noncooperative Behaviour and Escalation’, Journal of Conflict Resolution, 15:1, pp.109–11.
Page 115: The article cited on the ‘Winner’s Curse’ is Ed Capen, Bob Clapp and Bill Campbell, ‘Competitive Bidding in High-risk Situation’, Journal of Petroleum Technology, 23, pp. 641–53.
Chapter 10: Lies, Damned Lies, and Statistics
Page 136: Edward H Simpson’s 1951 paper ‘The Interpretation of Interaction in Contingency Tables’ was published in the Journal of the Royal Statistical Society, Series B 13, pp. 238–41.
Page 146: The two books I’m recommending here are: John Allen Paulos, A Mathematician Reads the Newspaper: Making Sense of the Numbers in the Headlines, Penguin, London, 1996; Basic Civitas Books, 2013; and Darrell Huff, How to Lie with Statistics, revised edition, Penguin, London, 1991.
Chapter 11: Against All Odds
Page 152: Stuart Sutherland’s 1992 book Irrationality: The Enemy Within was published in a 21st anniversary edition (with a foreword by Ben Goldacre) by Pinter & Martin, London, 2013.
Chapter 12: On Fairly Sharing a Burden
Page 157: The Airport Problem was first presented by S C Littlechild and G Owen in a 1973 paper, ‘A Simple Expression for the Shapely Value in a Special Case’, Management Science, 20:3, Theory Series. (Nov 1973), pp. 370–2.
Chapter 14: How to Gamble if You Must
Page 170: ‘Bold play’ is the phrase used in the Bible of roulette games: Lester E Dubbins and Leonard J Savage, How to Gamble if You Must, Dover Publications, New York, reprint edition, 2014; first published 1976 as Inequalities for Stochastic Processes.
BIBLIOGRAPHY
Chapter 1
Gneezy, Uri, Haruvy, Ernan and Yafe, Hadas, ‘The Inefficiency of Splitting the Bill’, Economic Journal, 114:495 (April 2004), pp. 265–80
Chapter 2
Aumann, Robert, The Blackmailer Paradox: Game Theory and Negotiations with Arab Countries, available at www.aish.com/jw/me/97755479.html
Chapter 3
Camerer, Colin, Behavioral Game Theory: Experiments in Strategic Interaction, Roundtable Series in Behavioral Economics, Princeton University Press, 2003
Chapter 4
Davis, Morton, Game Theory: A Nontechnical Introduction,
Dover Publications, reprint edition, 1997
Chapter 5
Gale, D and Shapley, L S, ‘College Admissions and the Stability of Marriage’, American Mathematical Monthly, 69 (1962), pp. 9–14
Intermezzo: The Gladiators Game
Kaminsky, K S, Luks, E M and Nelson, P I, ‘Strategy, Nontransitive Dominance and the Exponential Distribution’, Austral J Statist, 26 (1984), pp. 111–18
Chapter 6 & Chapter 9
Poundstone, William, Prisoner’s Dilemma, Anchor, reprint edition, 1993
Chapter 7
Sigmund, Karl,The Calculus of Selfishness, Princeton Series in Theoretical and Computational Biology, Princeton University Press, 2010
Intermezzo: The Raven Paradox
Hempel, C G, ‘Studies in the Logic of Confirmation’, Mind, 54 (1945), pp. 1–26
Chapter 8
Milgrom, Paul, Putting Auction Theory to Work (Churchill Lectures in Economics), Cambridge University Press, 2004
Chapter 10
Huff, Darrell, How to Lie with Statistics, W W Norton & Company, reissue edition, 1993
Chapter 11
Morin, David J, Probability: For the Enthusiastic Beginner, CreateSpace Independent Publishing Platform, 2016
Chapter 12
Littlechild, S C and Owen, G, ‘A Simple Expression for the Shapely Value in a Special Case’, Management Science 20:3 (1973), pp. 370–2
Chapter 13
Basu, Kaushik, ‘The Traveler’s Dilemma’, Scientific American, June 2007
Chapter 14
Dubins, Lester E and Savage, Leonard J, How to Gamble If You
Must: Inequalities for Stochastic Processes, Dover Publications, reprint edition, 2014
Karlin, Anna R and Peres, Yuval, ‘Game Theory Alive’, American Mathematical Society (2017)
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