Abstract: In this post we introduce the concept of Nash equilibrium, a fundamental concept in game theory. We then apply this idea to a small decision in the game Twilight Imperium. Following this, a discussion on how equilibrium fit as a solution concept and how the relate to game metas.
Over the past year, I have gotten quite into Twilight Imperium. My roommates and I played quite a number of games, and I also dipped my toes into the competitive scene. I have wanted to start writing more pieces for this blog, but didn’t really have any great ideas, until I saw something interesting happening in competitive TI play, specifically around the Trade card. Being the armature game theorist, I thought it best to analyze this Twilight Imperium is a legend among hobby gamers. The game comes in a box that could probobly fit a mid sized game collection, and takes a whole day to play! Taken from its BGG page, “Twilight Imperium is a game of galactic conquest in which three to six players take on the role of one of seventeen factions vying for galactic domination through military might, political maneuvering, and economic bargaining.”
As fun as it would be to take a deep dive on an algorithmic/mathematical analysis of the fun game, doing so would be impossible, as the rules are the size of a medium length academic paper. Thus in order to make any rigorous mathematical analysis, we will need to focus in on a small decision made during game play. Specifically, we are going to focus in on the specifics of the “Trade” strategy card.
First, a bit of background. One of the main systems in Twilight Imperium is the strategy card system. At the start of each round of the game, players take turns drafting a strategy card which gives them access to a extremely powerful action, once during that round of play. The “Trade” card, which is the focus of our analysis, can be seen on the right. All strategy cards have two different abilities, a primary ability, which is given to the player who drafted the card, and a secondary, which the other players are given the option off using (for a cost). Fully explaining this card requires a lot of other background knowledge, so (like any good applied mathematician) we will abstract them away, and focus in on a smaller detail, that being the final ability of the card.
Without knowing anything about TI4, it may seem like choosing to use the final ability on the trade card is a bad idea, as you are essentially giving players free “commodities” (Just think of it as some sort of money, the more the better). Why would that be a good thing? One of the primary mechanics in TI is the ability to trade with the other players at the table. Thus, players will often trade some of their gained commodities in exchange for getting their commodities refilled. With all of this is mind I present the following abstracted mathematical game, I call Give and Take!
The game works in the following way, first player A chooses any subset of players and offers them the following decision. They can gain three “commodities” but must give one of them two player A. Essentially the player gains two commodities, and player A gains 1. After, the players are ranked in order of who has the most commodities, and given utility equal to there position. (Essentially, players want to have more commodities than the other players) Now, before analyzing this game, I want to talk about the concept of equilibrium!
The concept of equilibrium is one of the fundamental concepts in game theory. Essentially, its a solution concept for a game, or a way of figuring out the expected/optimal outcome for a game. Formally, let be the set of all possible strategies (think decisions) for player , where . Let be a strategy profile, a set consisting of one strategy for each player, where denotes the strategies of all the players except i. Let be player i’s payoff as a function of the strategies. The strategy profile is a Nash equilibrium if .
In non math terms, an equilibrium is a state in game in which, no player is incentivized to deviate from their current strategy, assuming that all the other players don’t deviate from their strategy. Note that more than one Nash equilibrium can exist! Lets look at the example given on the right. There are two Nash equilibrium in this game, (Up, Left) and (Right, Down). The (Up, Left) equilibrium should be obvious, as it is the outcome that maximizes the utility given to both players, thus neither player wants to change which strategy they are playing. However, (Right, Down) is also an equilibrium, as Player 1, would receive no utility, if they switched to up. The same can also be said for Player 2, as if they switch their strategy from Right to Left, they receive less than they would have if they didn’t deviate. Thus (Right, Down) is also a Nash!
In competitive Twilight Imperium, the standard for Trade is for player A (the player with trade) to offer the deal to everyone and take it. In a standard 6 player game, this gives Player A a total commodity value of 5, and all other players a commodity value of 2. Thus Player A is the winner! However, couldn’t all player simply not take the deal? That would result in everyone having no commodities, which is a better outcome for the other players? Lets think this out.
Let’s first start and see if the situation in which all players are offered the deal, but none except. Is this in equilibrium? Surprisingly, the answer is no! As assuming all players keep the same strategy, a single player (let’s call them B) could choose to take deal, gaining 2 commodities, making the rankings B, A, everyone else. Thus this is not an equilibrium and now what we would expect to see in practice.
What about the behavior seen in the competitive Twilight Imperium games? Well, for each of the non A players, they are already making the best decision for them. If a player (B) chooses not to take the deal then they get 0 commodities, which puts them in last place, as everyone else has already taken the deal. Thus this is not a net gain! So for these players, they are in equilibrium. As for player A, they are already in the top spot, so they have no incentive to deviate. Thus, this outcome is indeed a Nash equilibrium! As such, although the outcome is worse for the other players, we should expect them to take this strategy, which it seems like they do.
I want to close this post with a quick discussion on meta and equilibrium in games. Often times in competitive gaming scenes we here this term “meta”. Essentially the meta is the most dominant strategy at a given time and playing outside the meta is generally scene as a bad idea. While the two are fundamentally different things, one being a rigorously defined mathematical construct and the other isn’t, I do think there is an interesting connection. We can think of the meta as a version of an equilibrium, in which players who deviate from it are worse off. Now, obviously this isn’t always the case, sometimes people have a wrong read on the meta and other strategies end up becoming more popular, but I do think it is interesting to note that the behavior we see in our models does have a pretty close match with what we see in actual competitive game situations.