Game theory is fun to study and it’s useful, but the reason that it’s useful is usually not because calculating and implementing a game-theoretically optimal solution is the best choice in a given situation. Often you’re better off behaving exploitably, and often the optimal solution is too difficult to discover, anyway.
Indeed, I’m tempted to say that I almost never have tried to implement a game-theoretically optimal solution to anything. I have dim memories of using random numbers to guarantee that a random decision was really random enough, but that’s about it. Most of my game theory studies have been for poker, where I could either only approximate solutions to real-life games (and then usually deviate from them) or else find exact solutions to toy games (that I would never actually be playing for money).
There was a time, though, when I found myself in an unusual spot — when I did both calculate and implement an optimal solution using game theory. And it wasn’t for making a decision at the poker table, but for picking the winner of a football game. However, the experience highlighted some of the reasons why studying game theory can be useful to poker players.
Let me explain.
Predicting Games… and Opponents’ Predictions
The situation arose for me at the end of last year’s college football season while I was playing ESPN’s “College Bowl Mania” game in a four-person group. The game had us picking the winners of all 35 college bowl games and assigning a number from 1 to 35 to each of our picks. If you picked a game correctly, you received the number of points you assigned to that game.
The twist was you could swap numbers among games and also change your picks, as long as the games in question hadn’t yet started. Some used that option to react to injuries and weather changes, but I soon realized that by far the biggest benefit of this ability to change your numbers was to adjust for your standings in your group.
Since we were playing to get higher scores than a particular set of people (and not just to maximize our own scores), it would sometimes make sense to switch a game I was less confident in to a higher-value slot or to pick the underdog in a game. These situations came up when I was losing and needed to give myself a chance to pick up points that my opponents wouldn’t be getting.
One Game, One Winner, One Champion
A relatively simple example of this kind of spot happened to be exactly the situation I found myself in the night before the championship between Florida State and Auburn.
Only two of us were still in the running to win the pool — the rest of the pool could not possibly catch up. I was beating my remaining opponent by roughly 10 points. We had each assigned the Florida State-Auburn game a value of somewhere between 30 and 35, and because it was the only game left, neither of us could change our number for the game.
So what did that mean? Well, there were two possible outcomes:
- If we both chose the same team as our pick to win the game, I would win the pool no matter what happened. That’s because we would both add thirty-odd points to our scores, or we would both add zero points to our scores; in either case, my 10-point lead would be enough.
- If we chose different winners for the game, whichever one of us who had chosen the winner would win the pool.
As I say, I couldn’t change the value of my pick, but I did have the option of changing my team choice as winner for the game right up until kickoff that night. I had picked FSU to win originally, but I could still change to Auburn.
What should I do in this situation?
Well, if I were confident that my opponent wasn’t paying attention, I should almost certainly pick FSU and be happy about it. Most people were picking FSU over Auburn — indeed, that was probably even more true of those who assigned a very high number to the game. Knowing that, I should guess my opponent would pick FSU, too, and thus by picking FSU myself I’d secure the win in the pool.
However, if I were confident that my opponent was confident that I would pick FSU, then I should switch my pick to Auburn, because he would be picking Auburn as it was the only way for him to win if I took FSU. In such a spot many opponents might figure that you would never switch away from FSU. If they assume that, though, you have an immediate win by switching.
I wasn’t in either of these situations, though. I only barely knew my opponent. All I knew about him is that he had caught up from a bad start in the competition by picking several underdogs — or, perhaps, he hadn’t ever bothered figuring out who was favored to begin with when he had made his previous picks.
Thus I wasn’t confident about anything when it came to what he might do. Maybe he would simply go for the underdog again. Or maybe he would try to outwit me by picking the favorite. Or maybe he was busy with his job and wouldn’t have time to switch from FSU to Auburn even if he wanted to.
I simply didn’t know.
Calculating an Optimal Solution… and Implementing It
Paper-scissors-rock was never my best game, so I decided to use a game-theoretic solution. Thankfully, I’ve read The Mathematics of Poker by Bill Chen and Jerrod Ankenman, so I knew how to calculate it.
I knew that my optimal solution would be the one that would make my opponent indifferent between picking FSU and picking Auburn. Let p be the probability with which I pick FSU, and let w be the probability that FSU wins. Then my opponent’s winning chances are:
(w) * (1 - p), if he picks FSU. (He wins when FSU wins and I pick Auburn.)
(1 - w) * (p), if he picks Auburn. (He wins when Auburn wins and I pick Auburn.)
w * (1 - p) = (1 - w) * p
w - wp = p - wp
p = w
Remember p stands for the probability that I pick Florida State while w stands for the probability Florida State wins the game. According to my game-theoretic calculation, I found that I should pick FSU exactly as often as FSU wins.
Before the game began, FSU was roughly 77% to win — or at least that’s what betting markets told me. So that meant I should pick FSU roughly 77% of the time, too.
I fired up something called a “Python interpreter” and had it generate a random number between 0 and 1. I resolved to pick Auburn only if the random number it produced was something greater than .77 — that way I’d be making the probability of my picking FSU exactly match the probability of FSU winning the game.
The random number that was generated was greater than .77. So I took a deep breath, switched my pick from FSU to Auburn, and waited.
The Game Begins… and My Game Ends
While I waited for the game to begin, I had time to reflect on the strangeness of the situation. It’s rarely feasible to find an optimal solution in a real-life situation, and even when it is, it’s rarely correct to use it.
I was reminded of the winning strategy Benjamin Morris employed in ESPN’s Stat Geek Smackdown a couple of years ago. Morris faced a structurally identical situation to mine, but he was in my opponent’s spot. He simply figured his opponent, Stephen Ilardi, would never switch to the underdog, and that assumption served him well.
Kickoff brought good news: my opponent had taken Auburn also! I couldn’t lose!
Maybe he had taken Auburn figuring I’d always take FSU. Maybe he simply thought Auburn would win. Or maybe he had done the same calculation I had and used a random-number generator to choose Auburn 77% of the time and FSU 23% of the time.
How an Understanding of Game Theory Can Help Poker Players
There are, I think, a few lessons here for poker players.
One is that, if you’re serious about the game, you will spend lots of time thinking about situations that are only approximations or simplifications of those you will actually face on the felt.
I’ve solved toy games and worked through a lot of simplified EV calculations, and game conditions are always more complicated. Even jam-or-fold decisions, for which we know a lot about optimal play, are not so simple in live games where you often have additional information about your opponent’s suboptimal tendencies or his physical behavior.
But it’s a mistake to think that these toy games or other simplified EV calculations aren’t relevant just because they don’t exactly match in-game situations or because your opponents are exploitably weak. Real poker decisions are often related so closely to those simplifications that a good foundation in the latter will help you immensely. And, as I discovered in the football pool, sometimes situations will fall into your lap where you do want to revert to optimal play and it is computationally tractable to determine what that play is.
A psychological trap lurks on the other end of the spectrum, however. Players with excellent fundamental knowledge often rely too much on the situations they’ve solved and don’t look hard enough for information that complicates a given situation but also makes better plays possible.
A final lesson is that the structures of real situations, both in and out of poker, are often entirely different from the structures they appear to have.
The football pool I was playing in for fun appeared to be a test of predicting football. Actually, anyone with a basic knowledge of football and a good working understanding of game theory would be a huge favorite over the best sports handicapper in the world in “College Bowl Mania” if that handicapper didn’t also know when to pick the underdog and switch point values among games.
Compare this with poker, which many beginners think is a game of trying to make pairs and flushes. Anyone who thinks this way is dead in the water against someone who understands how to construct ranges and think about the game more abstractly. Fundamental but essential achievements in a beginner’s poker studies include being able to learn to see the game as sets of possibilities to be navigated, to think about counterfactual possibilities as essential parts of one’s reasoning about the present one, and to think in terms of long-term win rates instead of who won the last $100 pot.
This, in turn, points to a reason why poker can be so valuable to learn. When you come to see poker in such radically transformed terms, you heighten your ability to see other parts of the world more accurately, and you also heighten your skepticism about whether those parts of the world really are as they appear. In this way, among many others, poker can be a valuable part of your education.