Following my last post about game theory, I will dig deeper in this post about a type of game that in my opinion, is very common. Asymmetrical games root themselves in real-life situations where the conditions vary greatly and more often than not, two opponents would find themselves on unequal grounds, with each having different strengths, weaknesses, and even motives. A thorough study of this type of game, therefore, proves to be an efficient tool for strategists to calculate the chances of success for different, and more often than not, hardly related players and gives the optimal strategies ranging from pursuing specific targets to negotiating for a potential win-win situation.
Firstly, we would go through the definition of an asymmetric game. Although the definition varies, most of them often carry this phrase or a paraphrase version of it:
“Asymmetric games are games where players have different, and even conflicting, goals and strategies”
So much is obvious, yet its implications are massive. Although most of the games that are modeled into game theory examples are symmetric games because of their elegance and straightforwardness in strategy-planning, in real life those situations do not often occur. Instead, the diversity in personalities and traits between humans adds so many variables into the picture that the same strategies can hardly work for every single individual. Moreover, humans also possess a rational brain that constantly adapts to the situation and devises appropriate goals and strategies, making symmetric games even more short-termed. In the end, we obtain a truly chaotic reality that can only be best represented by a model of an asymmetric game.
Take a simple example of the stock market – a platform that is constantly updated and operated by millions of investors. Every minute, thousands sell their stocks while others buy in, waiting for a potential rise in the stock’s price. In this example, although the end goals are similar – to maximize profits – the strategies are drastically different, from the invested company to the amount that people buy or sell. With a closer look, the goals can be divided into different subcategories too, like long-term investments, short-term investments, or simply to avoid further losses. This diverse “player base” is responsible for the extremely messy environment of the stock market and hence its nature as an asymmetrical game.
From this example, we could observe a quite important property of an asymmetrical game. An asymmetric game is likely, though not always, multi-sided, and it is easy to understand why: the more people join the game, the greater the diversity in terms of traits. More often, a game with just 2 players can boil down to fair competition, but asking for such a thing from many players is just too much.
Another noticeable trait for the game is the fact that it often either has many goals or has a goal that can be interpreted in different ways by different players. An online RPG game, for example, is asymmetrical because players can strive for many different targets. A stock market represents the other side: it has a goal – making a profit – that can be interpreted differently by different investors with their resources. On the other hand, asymmetric games like tic-tac-toe have one goal – to win – and one method – to build a string of symbols with predefined length, proving the extent the trait is associated with asymmetric games.
In the end, I would like to present the most interesting example of this type of game, and that is the fight for survival. In nature, the relationship between the predator and the prey makes it a perfect example for an asymmetric game, with clearly differentiated goals – one is to hunt, and one is to escape – and various tactics to adapt to these goals that have for so long captured the attention of animal behavior researchers. In the IMO 2017 contest, Problem 3 illustrates this game using the view of combinatorial geometry and creates a perfect material for thought:
A hunter and an invisible rabbit play a game in the Euclidean plane. The rabbit’s starting point, , and the hunter’s starting point, , are the same. After rounds of the game, the rabbit is at point and the hunter is at point . In the nth round of the game, three things occur in order.
(i) The rabbit moves invisibly to a point such that the distance between and is exactly 1.
(ii) A tracking device reports a point to the hunter. The only guarantee provided by the tracking device is that the distance between and is at most 1.
(iii) The hunter moves visibly to a point such that the distance between and is exactly 1.
Is it always possible, no matter how the rabbit moves, and no matter what points are reported by the tracking device, for the hunter to choose her moves so that after rounds she can ensure that the distance between her and the rabbit is at most 100?
For now, math on!
Crash on Maths 