This post we’ll be talking about games—contrary to what people often say in dramas, this is a game.
An extensive-form game is a tree of decisions branching out, with actors forming the nodes in the branches, and the branches representing choices that the actors can make. The assumption is always that each actor is making rational choices, trying to get their best outcome, at every point.
To determine the outcome of an extensive-form game, you work from the ends backward to the beginning, using backwards induction. To demonstrate, here’s this game:
In this game, the actors are a follower (F) and a leader (L). The follower acts first, and has the choice to stay home or volunteer. If he stays home, the game is over. If he volunteers, then the leader has a choice. She can honor her follower or she can exploit him. Of course, since these actors are rational, they have preferences. The preferences are noted in the parentheses in this order: (Follower, Leader). So the follower values the stay home outcome at 10, the honor outcome at 50, and the exploit outcome at 0. The leader values the stay home outcome at 10, the honor outcome at 50, and the exploit outcome at 51. Now, it might seem smart to start from the beginning, but it’s not. The follower’s decision depends on what he thinks the leader will do, and the leader’s decision depends on what her preferences are, so let’s start there.
Completely ignoring the first numbers in those ordered pairs, the leader has the choice between an outcome of 50 and an outcome of 51. Of course, she’s going to choose the 51, because it’s higher.
You might say, but doesn’t she care about the follower? No. If she did, she wouldn’t value the honor outcome higher. You might also say, but wouldn’t it be best if she promised to honor him and then did, so that he trusted her the next time? Yes, but this is just a single game. Once it’s over, it’s over. The world starts at the beginning of this game, and ends at the end—which is why games are particularly good for war, when the end of the game really is the end of everything.
So the leader is going to choose to exploit her follower. The follower knows this because he knows what her preferences are, and he knows what she’ll choose. So the follower has the option of staying home, which will give him an outcome of 10, or volunteering, which will give him an outcome of 0. So of course, he’s staying home. And that’s our outcome—stay home, (10, 10).
This backwards induction works for any game with complete and perfect information.
Complete—all rules of the game (all the possible choices) are known. Perfect—everyone observes all past choices. So games like tic-tac-toe, connect-four, checkers, and chess can be solved with backwards induction. All of those games have been solved, except chess.
By contrast, rock-paper-scissors can’t be solved because it does not have perfect information—you don’t know the other player’s choice until after you’ve both gone.
Calvinball does not have complete information (and probably not perfect information either, but I guess it depends.)
This is a limitation of extensive-form games—they require the game to be played in a sequence, with each player taking one turn at a time. But not all games work like that, as exampled above. To model RPS, we would need to use a normal-form game.
The outcome of a normal-form game depends on the simultaneous choices (or at least independent choices, each made without knowledge of the other player’s choice) of each player. So how do you win such a game? That depends on expectations. If you expect someone to play scissors, you’ll get your best outcome by playing rock.
Now, let’s bring back dominance theory. From that post, you may assume that it is always best to be dominant—but maybe not.
An example involving a dominant and a submissive pig:
These two pigs are in a pen, and to get food they have to push a lever. The lever will release 10 carrots into a trough at the other end of the pen from the lever. The lever takes a lot of effort to push—let’s say 1 carrot’s worth of energy. So best to wait by the trough while the other guy pushes the lever, right? Here it is as a normal-form game.
The dominant pig’s outcomes are on the listed first (so the numbers go dominant, submissive.)
In this model, the actions of the submissive pig are shown as columns, and the actions of the dominant pig are shown as rows. The outcomes are shown for every possible pairing of actions—the bottom row shows all the possible outcomes of the dominant pig waiting, and the bottom right space shows the outcome of both pigs waiting.
As with an extensive-form game, actors act based on what they anticipate the other actor will do. So we need to find out what these pigs will do for every possible action that the other could perform. This is a simple one, so it won’t take long. Let’s start with the submissive pig, because it works out cleaner that way.
If the submissive pig expects the dominant pig to push the lever, he can push the lever or wait. If he pushes the lever, he’ll lose 1 carrot of energy, and the dominant pig will beat him to the trough and hog all the carrots. So he’ll get -1 if he pushes the lever. If he waits by the trough, he’ll be able to eat as many carrots as he can (5 in this example) before the dominant pig makes it from one end of the pen to the other, and bashes him out of the way eats the rest of the carrots. So the submissive pig will have an outcome of 5. His choice is between -1 and 5, so of course he’ll choose to wait.
If the submissive pig expects the dominant pig to wait, he can either push the lever and wind up in pretty much the same situation as above (with a -1 outcome), or he can wait, and get no food for an outcome of 0. So he’ll choose to wait, and take his 0 instead of -1.
It turns out that no matter what the submissive pig expects the dominant pig to do, it’s always better for him to wait. So the dominant pig should expect the submissive pig to wait. Given this, the dominant pig can push the lever, and get 4 carrots (5 carrots -1 carrot of energy for pushing the lever), or wait and get 0 carrots. So the dominant pig should push the lever to get his preferred outcome of 4—and that’s the solution. Submissive pig: 5, dominant pig: 4. In this game, it’s actually not good to be dominant, because of the power of expectations.
So next post (the final post in this series on political analysis and power), I’ll talk about expectations, and how many divisions the pope has.
“I know, I’m having a crisis of why has it not ended.”