Games & Theory
Bernhard von Stengel is professor of mathematics at the London School of Economics. He sits in an oddly shaped corner office at the Columbia House and plays puzzles. By “playing” I mean trying to draw insights that he can later teach his computer to solve them more effectively. That’s kind of his job.
We met after corresponding for a while about the connection between Game Theory and the social interactions we call games. We talked about duopolies and prisoners in hats and how to make deductions like a true mastermind. This followed a series of interview questions I have sent him and he was kind enough to reply
You are a game theorist…
- What is your favorite game (Poker, Monopoly, Hide-and-Seek, Tic-tac-toe, Super Mario World, World of Warcraft, etc.) and why?
I play games alone for distraction, and these are puzzles that take a few minutes or, when interesting, 15 minutes. For a while I played the brilliantly designed puzzle 2048 where you move tiles with numbers on a 4×4 board where tiles with equal numbers merge, so two tiles with number 2 merge into a 4, and so on. After each move a new tile with number 2 or 4 appears in a random place, and the goal is to reach a tile with 2048 (or higher) before the board is full. What makes this puzzle so addictive is the unlimited speed at which you can move the tiles once you have a strategy. I get to 2048 in 5–10 minutes, and three times made it to 8192 which is probably an hour’s play. Of course, you never win, other than by improving your high score, and at some point have to stop playing this puzzle.
Two more puzzles I play regularly on the computer. One is an old one called Mastermind, where you have to guess a “code” of 4 pegs each of which can have 6 colours. You propose a code and get partial information on your guess in the form of black pegs (colour and position match, but not where) and white pegs (other pegs of correct colour but not correct position). In theory, you can always succeed in 5 guesses (and on average a bit over 4 guesses), but the optimal strategy is impossible to remember by a human. So I use some self-found heuristics and am pretty good. I play each round in about a minute and it increases my concentration. The only boring thing is that I cannot much improve my strategy any more.
I play Mastermind (which was invented and sold in the 1970’s with plastic pegs) in a very nicely minimalist design under the name “Guess” in Simon Tatham’s Portable Puzzle Collection which can be found for portable devices under http://hewgill.com/puzzles/ (I play it on my Linux-powered laptop) and this puzzle collection has a second favourite of mine called “Galaxies”. I choose a 15×15 board under the (custom) “Unreasonable” option. It displays, always varying, on a board with 15×15 squares a bunch of white circles around which you have to create a rotationally symmetric “galaxy” of squares (e.g., a 2×2 square with the circle in the middle). The entire board has a unique tiling into such galaxies which you have to find. This is visually, geometrically, logically challenging, and very satisfying when found. I can complete
it between 4 and 15 minutes. I am thinking of improving the (open-source) code so that a single click on a circle expands (or shrinks) to a maximal (or minimal) galaxy which would improve the speed of the game. At the moment you spend a lot of time filling the board with a first tiling that you then try to improve. Maybe I should do this rather than playing the game, although that programming job would take many hours. I also should do research or writing or reviewing papers rather than doing this, of course.
I also solved myself the Rubik’s cube and have a self-found set of operations to bring it slowly but surely to a solution in about 3 minutes. (I can hardly leave alone a scrambled Rubik’s cube that I find without itching to solve it.) Now that I have a way, I am not ambitious enough to make that faster. The fun is in devising your own solution method.
1b. Games with others
I play games for fun with others. Although I sometimes take them too seriously, I think a game should not be a science, which defeats its social purpose.
I play Chess occasionally with my son. Rapidly and not very well, with attempts at all-out attacks and with taking back foolish moves, in agreement. You can ruin a whole chess game with one bad move, which is my main objection against this game for entertainment purposes. However, Chess is very well designed in its complexity and balance of attack and defense. I go almost entirely by “feel” of the geometric arrangement of the pieces, with very little think-ahead capability. If I wanted to play Chess well I would have to study it seriously and see this as a dead end, for obvious reasons. Better learn the piano, which I also don’t do.
As concerns games with many players, my current favourite is Resistance, a role-play game which you can play well with 6 to 8 players and with minimal equipment (a few playing cards). I like the simplicity of the original version, and most of it the social aspect where you have to convince and deceive others. You also have to be careful about your preconceptions of who is on your side or not.
There is a card game in Germany called Doppelkopf that we played a lot in our family (with my parents and my brother) for years every Sunday. It is a trump game like Bridge, and for that reason hard to explain to any new player who has never played a trump game. It is a derivative of the three-player card game Skat, very popular in Germany. Unlike Skat and Bridge which have become scientific in the way they have become analysed (in other words, there is an optimal play), Doppelkopf is more playful. It is also simpler in that there is no bidding round, and therefore faster — very important for me because I am impatient. The main idea is that you have 4 players who in each round are different partners of 2 pairs determined by the cards, and you have to find that out during play. Similar to Skat, the high trumps which get you tricks do not count much, so you have to find ways of collecting points with the “fat” but less powerful cards. There is a lot of signaling going on, with conventions, and random elements that make mistakes less onerous. I tend to play it very “rationally”, i.e. help my partner only if I know who that is, but others also gamble more.
I was once on an academic workshop where we played Doppelkopf (and was admonished by my PhD not to take it so seriously) and found something very interesting after playing some rounds of Poker before: namely, I was trying to read people’s faces to find out what cards they have, and if they are on my side or not — something you are not meant to do, but which comes naturally with Poker where that information is paramount.
In short, I like card games because of their hidden information and how to elicit it. Further things in favour of card games are their speed and the fact that mistakes are local.
There is one less-known board game that I would like to recommend, called Wealth of Nations.
It is played with 3 or more players, and (unfortunately) takes about 2 to 3 hours to complete. On a central board with hexagonal tiles, you can place tiles of various “industries”, such as agriculture, mining, manufacturing, education, energy, banks, which have economies of scale because adjacent tiles create additional production opportunities (cleverly designed in the form of matching half-circles along an edge of a tile). In each round you produce goods with your chosen industry, and those goods have prices according to a cleverly represented supply and demand mechanism (little blocks that are added or removed from a scale that indicates the price). So you should be in an industry that is not also provided by others. This supply and demand, in addition to where you put your tiles, is the main interactive aspect of the game. Much of the time you are occupied with planning for your own economy, how much debt to take on (again with a clever simple mechanism), and so on. So at the end of the game even the losers feel good because they have developed their own system. The game is also not completely predictable because different industries pay off differently over time.
This unpredictability is very much in contrast to other long board games which become boring half-way through the game because of the “runaway winner” effect where you can already tell who is winning way before the game is finished, as in Settlers of Catan, Risk, or most extremely, Monopoly. These long games are not worth their length of play.
A light-hearted well-designed game, much shorter, is Carcassonne, which one can also play well with kids. Very well designed because the cards that you lay always look different.
- What attracted you to game theory as your field of research?
This is of course a different question from playing games.
I studied mathematics and was slightly frustrated in learning about the edifice of mathematics that I thought I could never contribute to, given all that hard stuff that was already solved in the 19th century, most of which was beyond my comprehension. I was therefore also all the time halfway into computing and programming which was much more creative and hands-on in comparison. But then, after my basic studies (which I did well in) I took a course on Decision and Game Theory that was an eye-opener. It taught about decision theory with some basic insights that I found very useful as an attitude to life: If you face a risky decision and it comes out bad, that is fine because you were aware that you took a risk. Similarly, if it came out good it does not mean it was the right decision. Simple recent political example: I found the referendum on Brexit a completely irresponsible gamble by British prime minister David Cameron the moment it was made, even if it had not come out the bad way it did.
I then worked on decision theory in my MSc and PhD, because it was relatively new, and I found many recent papers which I could understand mathematically, and improve, and find new connections in. Decision theory and game theory offered many questions where you could contribute, as a mathematician, without studying hard for three years before you even understand what the questions are. So, as a research subject, game theory is accessible mathematically, and offers a lot of opportunities to find nice and elegant questions from a mathematical viewpoint.
- Do you get to play many games in your spare time? Did you used to, when you were younger?
I play probably too many of these puzzles, in between, see above, and occasionally games such as resistance when in a group (but where we mostly prefer to talk so we don’t play that much). As a kid, I played every Sunday afternoon with my parents. In addition to Doppelkopf also Canasta where you have to think less. So these were weekly sessions of 2 to 3 hours, for about 5 years when I was aged 12–17 (and my brother 7–12).
- Do you actually play games as part of your studies? And I don’t mean through computer-executed algorithms.
I do NOT play games as part of my studies, but I think the cognitive aspect of how to play them is interesting (already in the very restricted puzzle Mastermind). I am not into studying that yet but I might.
Games in game theory is a well-defined mathematical objects, whereas “actual” games, as we non-game-theorists know them, don’t always strictly fall under that category.
The name “game” means that you have rules, and they should be clear and understandable. Most board and card games are very precise in that regard. Whether they are fun to play is a very different matter.
- Can you give an example of an “actual” game you played, such as a tabletop game, video game or party game, where you clearly noticed game theory come into practice? Did you win?
I was once at a summer school with hyperintelligent students where we played Mafia (a precursor of Resistance) and in the final round there was a semi-obvious move of whom I would protect, and an opponent out-smarted me by anticipating this move, convinced the others to act accordingly, and so I lost. I was not good enough at “I think that you think that I think” which I should have been as a game theorist.
- Have you solved any game theory problems, or researched any game theory principles, that you think can make for a good “actual” game?
I have not tried to design a game, but my son, now 17, has designed games for 10 or more years. (He thinks my puzzles are boring.) A game should be simple, fast, work on many levels, and have good tension-and-release cycles. Among my colleagues, I get computer scientists much easier into play than game theorists. So, no, no games from game theory.
Good games shouldn’t have a winning strategy built into them, but winning strategies (or rather “equilibria”) is exactly what game theory is trying to figure out.
- Does game theory ever deal with games that have no winning strategies, such as games involving uncertainty or games with more abstract goals? If so — how can it be applied then?
Game theory does deal with games that have uncertainty, such as Poker, where “winning” works only on average, but the theory applies. In game design, having fun and feeling good after a game are goals that are not modeled by game theory.
- What can game theory say about creative “moves” that may defy logic?
A Poker player who can read his opponents’ faces is not using logic but other insights (things we humans are actually extremely good at an intuitive level). Game theory is very reductionist in that respect.
Speaking of logic, or lack thereof, game theory is defined for intelligent rational decision-makers. However, behavioral sciences have continuously proven that people can be very irrational, and that their intelligence is questionable in certain circumstances…
There has been some “theory” that reading people’s intentions (to deceive, for example) has been evolutionarily very important for humans, and contributed to the success of societies as a means to cooperate. This seems to me common sense, maybe a bit of psychology, but not a deep insight from game theory. In fact, many insights from game theory are actually common sense — the surprising bits are much rarer.
- Does game theory have models for non-rational players, namely humans?
There is a lot in game theory about bounded rationality. Whether this models adequately is another matter.
I agree to a large extent, although not fully, with Ariel Rubinstein, see e.g.
- Can game theory help assign human ingenuity to computers? Should it?
There is a lot going on about machine learning and artificial intelligence at the moment. The human ingenuity required for rather mundane tasks is still enormous and poorly understood. I think the interactive part, which is what game theory has to offer, will come at a later stage.
While game theory deals with abstract models of systems and their elements, these can describe “real world” situations for “real” participants…
- For an ordinary person, what would be the best insights that game theory can provide? Are you using these insights in your own life?
Apart from what I said about risk above, the main message of game theory is that you should be aware that many situations are interactive and that it helps to put yourself into the mind of others. It also helps to think ahead.
One basic message of game theory was made cogently by Von Neumann: to complain that people are selfish is like complaining about gravity. So the Prisoner’s Dilemma is a basic insight that individually selfish and “rational” acts can result in non-cooperation that is detrimental to all. On the other hand, many societies have developed conventions of behaviour (such as not littering your streets, in Switzerland more so than in India) where cooperative behaviour seems to be self-enforcing. This behaviour is actually collectively more rational and beneficial than the apparently completely “selfish” model of game theory.
Game theory has been attacked for its “rational” and “utility-maximizing” approach. Both terms actually mean the same in standard game-theoretic assumptions. For the layperson, this is actually a typical misunderstanding of a precise and narrow mathematical term which is read with additional connotations that actors are cold optimizers.
For daily life, the main help of game theory or in fact of mathematics is that you should think rigorously and question your assumptions.
- For humankind and our civilization as a whole, what would be the best advice that game theorists can give us? Can it save the world?
See others as people like yourself. Is your argument still valid when you exchange roles? (If this would be applied sensibly then religious fanaticism should vanish, but unfortunately what is missing there is being sensible in the first place.) Think ahead. Incentives matter. The rules of the game matter.