The King’s Gambit: Zermelo’s Theorem and Quantifying Decision-Making in a Monopoly Market
While watching a GothamChess video on a game analysis of Garry Kasparov and Anatoly Karpov in the 1985 FIDE World Chess Championship, which is a game that is a pivotal point for two of the most revered chess players of all time. As I went on to most of the details of the game, if chess is a game with such heavy implications on decision-making and theorems, such as opening books, as an economics student, I began to wonder, how can chess theorem relates to economics? Browsing through the internet, I stumble upon what is called Zermelo’s Theorem.
Anatoly Karpov vs. Garry Kasparov, final board, round 24, 9th November 1985 (Source: Chess.com)
So what really is Zermelo’s Theorem? The theorem is about finite two-person games of perfect information in which the players move alternately and in which chance does not affect the decision-making process. The theorem is named after Professor Ernst Zermelo, a German mathematician and logician, in his paper “Über eine Anwendung der Mengenlehre auf die Theorie des Schachspiels” in 1913, proving the theorem and became one of the many theorems that are the backbone of game theory.
In order to satisfy the theorem, the game must satisfy the following criteria:
- There are two players in the game.
- The game is of perfect information.
- The board game is finite.
- There are three (or two outcomes)
- Win, Lose, and Tie.
Zermelo stated that either 1 can force a win (for 1), 1 can force a draw, or 2 can force a loss (on 1) and vice versa. Basically, to put it into words, Zermelo’s Theorem is a pure strategy for a player i in a game of perfect information, which is a complete plan of actions in which it specifies the action that i will be taken at each of its decision nodes.
The Foundational of a Theorem
When applied in a game of chess, the theorem states, “Either White can force a win, or Black can force a win, or both sides can force at least a draw.” Zermelo determines that a two-person game is without chance, in which players have opposing interests and a finite number of positions. From this, he addresses two problems. First, what does it mean to be in a winning position, and can it be defined in a mathematical equation? Second, if someone is in a winning position, can that number of moves force the win to be determined?
To answer the first question, Zermelo states that a certain set containing all potential moves such that a player wins regardless of how the other player plays is a necessary and sufficient condition. However, if this set is empty, the best a player may hope for is a draw. In order to allow a player to postpone his loss for an indefinite number of moves, Zermelo introduces another set that contains all conceivable move combinations, which implies a draw. This set might also be empty, meaning that if the opponent plays well, the player can only stop losing for a limited number of moves. However, this equates to the opponent having the ability to compel a victory.
On the second question, Zermelo argued that the game would never require more moves than there are possible positions. He supported this claim with a proof by contradiction: Suppose a player could win in more moves than there are positions in the game. According to the pigeonhole principle, at least one winning position must have occurred twice. Therefore, the player could have employed the same strategy when the position first occurred as he did when it appeared again, leading to a victory in fewer moves than there are positions available.
It Takes Two
Zermelo’s proof of this theorem is based on proof by induction, meaning there is a logical progression of justifiable steps in first asserting a hypothesis. So, to prove this, we have to look at the maximum length of a game. Suppose we call this game, N and in this game, players 1 and 2 take turns. In the picture below, we look at if the length of the game is n (e.g. N=n). In which, the decision nodes will look like this.
Decision Tree of a Zermelo’s Theorem Game N=n
(Open Yale Courses ECON 159: Game Theory Lecture 15)
So, by the induction hypothesis, there will be two subgames: an upper subgame, let’s say, W1 and a lower subgame, which is L1 and to conclude, that up is a winning strategy for player 1 and down is a losing strategy for player 1, and vice versa.
Taking Theory to Board
After that long lecture on logic and mathematics, you might be wondering how these bundles, branches, and theories actually applied to a real game. So, let’s take these theories and put them into a chessboard, specifically into an opening book, which is a set of moves in the initial part of the game, in which chess players are categorized into established theories. So, let’s take one of the opening theorems that was popular in Zermelo’s time: The King’s Gambit.
The king’s gambit is an opening book for White and, like any other gambit in chess, implores the player to sacrifice one of its pieces to gain a positional advantage. The move begins with the classic Pawn to e4 and Pawn to e5, but rather than Knight to g4, opening the Italian or Spanish opening, the Pawn moves to f4.
King’s Gambit Opening (chess.com)
So, how do we put this in branches such as the Zermelo theorem implores? Let’s say the moves are defined as e4, e5, and f4, just like the King’s Gambit line, and the game is defined as K, then the branches of the payoff of playing any other legal moves defined as x,y, and z would look like this.
King’s Gambit Zermelo Branch
Because Zermelo’s Theorem implores us to think in a backward induction, in this scenario, White has four strategies, which are [e4f4], [e4z], [xf4], and [xz], while the Black strategy is only e5 and y. By backward induction, we found that the best move would be ([x|z],y).
King’s Gambit Zermelo Matrix
How People Interact in a Monopoly Market
We’ve talked about math, chess, and math again, and yet we haven’t answered the first question: How does this theory and chess explain decision-making in economics? To answer this, we have to look into a monopoly market. Let us imagine a market in which in this market there is a monopolist, but there is someone, an entrant, that is trying to enter the market. This entrant has two choices: to fight or not to fight the incumbent monopolist. The Zermelo’s branch would look something like this.
Monopoly Market Zermelo’s Branch and Matrix
(Open Yale Courses ECON 159: Game Theory Lecture 15)
Seeing this, the Nash Equilibria would be (IN, NF) and (OUT, F). This means that if the incumbent chooses to fight the entrant, the best response would be out, and if it chooses not to fight, then the incumbent best response is to enter and become a duopoly market. How about when we analyze this from an incumbent point of view? If the incumbent gets to move, then they would choose not to fight, and if the incumbent chooses that, the entrant chooses to go in because it’s more profitable.
Conclusion
We must understand that the equilibria of Zermelo’s backward induction are used in believing in incredible threat. Meaning that we have assumptions that the players we are fighting against are going to make the move we believe them to happen. In the end, even Zermelo’s theorem is only a strategy. We never would predict in the chaos of this universe the next move of someone, whether it is in a chessboard or in a market. As Robert J. Oppenheimer said, “Theory will only take you so far.”
Imam Prabowo | Economics 2022 | Staff of Kanopi FEB UI Economics Studies Division 2023/2024
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