I have final exams next week so I’m using this post as a way to study, which is why it contains more definitions and a bit less original commentary.
My significant other and I have gotten into chess recently. We don’t know the names of the openings, the best ways to checkmate, and when we aren’t together and play others online we usually lose. But we know how the pieces move, are learning, and enjoy playing since we are both competitive in nature.
What I’ve learned so far is that you generally gain advantage by taking the opponents pieces. And the more valuable the piece the better. This seems obvious, but generally taking an “uncovered” piece is much more preferable than taking one that is “covered.” What I mean by this can be explained in the following diagram:
In this board black’s queen is uncovered since black is “in check.” They must either take the knight with a pawn or move the king to avoid check mate, leaving white to attack with their queen, without sacrificing their own queen. The black queen would be considered “covered” then if it there was another piece that could take the white queen should it choose to move to take the black queen.
How does this game relate to Game Theory?
If white were to take black’s queen, that would likely be a Best Response for white in the game. Now, it isn’t the only move they could make. For example they could take black’s knight with their rook. But since the queen can move in more ways than the knight, it is considered more powerful and a better move to take for white, especially since white doesn’t have to give up their queen.
The “Best Response” in game theory terms officially, is a strategy set that results in at least as much utility as every other strategy set, given the other player’s strategy. It is a strategy that maximizes the payoff conditional on all other players’ strategies. What a strategy set is, is a set of decisions in each state of the game. So if white opens with pawn to e4 (a very common chess opening) then black knows what to do in that case. Then after that, white moves another piece based on black’s move (which was part of their strategy set!)
So if white’s payoff is maximized when winning the game of chess, then white’s best response would likely be to take black’s queen. As I understand, assumptions such as this are what the various openings and end games are built on. But because there are so many combinations of moves it might not actually be the best response to take the queen, depending on if white anticipates correctly what black will do and if there is a way to check mate instead by not taking the queen.
If the goal is to check mate the opponent, it might be helpful to try to imagine what that would look like and then work backwards to position yourself to that point. No matter what the general strategy is, it likely wouldn’t involve leaving lots of your pieces uncovered, and exchanging your lower valued pieces (like pawns) for your higher valued pieces (like queen). Something like this might be called “Never a Best Response” or a strategy where you can find a better one conditional on all other players’ strategies.
If this is true, you would likely choose to eliminate those strategies, through a process called “Iteration of Never a Best Response.” And whatever strategies you are left with are called rationalizable.
If all players play a strategy that results in best responses to each other, then that is called a Nash Equilibrium. (You may have heard of this before, named after the famous economist and mathematician John Nash who is the focus of the biopic “A beautiful mind” starring Russell Crowe which is a great)
A Nash Equilibrium doesn’t say that this is the outcome that will happen. It also doesn’t say that it is the outcome that should happen. In fact, it’s not very helpful at all, especially since there could be many Nash Equilibria in a game and there is an underlying assumption that each player knows what their best response is, and that what they believe the other players strategy to be is correct.
But some Nash Equilibria are more helpful than others. A subgame perfect Nash Equilibrium is one that has a Nash Equilibrium in each of it’s subgames. This is more helpful because you might be able to anticipate what the other player will do, and act accordingly, flowing through the series of moves in a way where each player is choosing their best response to each move.
In chess, this would look like each move by each player is a best response to each move. Now, there are too many combinations of moves to do this full analysis, but I would suspect that this would result in either a draw or a win by white since they move first.
There are a lot of problems with game theory, namely the perfect information assumption, the assumption that magnitudes don’t matter (only the inequality between payoffs, and that the results don’t model human behavior. But it can be a useful framework to think through a contested decision problem.
How about another example
One game I’d like to talk about is called the ultimatum game. In this game there are two players, and they are attempting to divide a pile of money. Player 1 can divide the pile any way that they like, and Player 2 can choose to accept or reject. If player 2 rejects, they both end up with 0 money, but if player 2 accepts, they both get to keep the pile of money. Say player 1 can’t offer 0 dollars, then depending on the amount of money is in the pile then player 1 can have that many strategies to offer to player 2. Then, player 2 can have 2^that many strategies (need to choose yes or no for each of them). The only subgame perfect Nash Equilibrium for this game is the one where player 1 offers $1 and player 2 accepts no matter what.
The best response for player 2 is to accept anything, since some money is better than no money, and then given this strategy player 1 could do no better than only offering player 2 $1, and so this is the best response for each player, and the best response given all of the subgames.
This is a bit unexpected! If I’m player 2 of course I would rather have $1 than $0, but if the pile of money was $10,000 why should I let player 1 have that whole pot? What did they do to deserve it? Why not split it evenly wouldn’t that sound more fair?
When the game is played in a laboratory setting this is generally what is found, if not a 60-40 split. So what is the point in talking about this game? To me, this game sounds a bit like a salary negotiation. The employer offers the potential employee a salary, and since the potential employee in unemployed earning $0, they would rather have some money than no money, and if they don’t end up taking the smallest allowed amount (here it would be minimum wage) they would end up being underpaid.
And it’s not just me who thinks this. Economists at University of Reading reproduced the 60-40 split result in a salary negotiation setting, and found that when employees were asked to complete a task beforehand they were likely to be paid more.
This makes sense to me, during a salary negotiation there are inherently information asymmetries. Both on the employee quality and the employer quality and position budget. But I do suspect this game could suggest that employees are underpaid.
*Update* I hadn't seen this originally but thanks to the reader who pointed this out that the black queen is technically not uncovered yet since if the black king moves out of check diagonally (next to rook and knight) then the queen would be covered then. Point being that you think your best response is to try to capture uncovered pieces to gain an advantage in the game.