Games and Economies
On Games, Part 3: Perfect Information Games
In this article, I talk about perfect information games in the context of the market economy framework from Part 1 of this series.
The economics literature distinguishes the quality of a game’s information (perfect vs. imperfect) from the completeness of a game’s information (complete vs. incomplete). Perfect information means that every player is perfectly informed of all the actions that happened in the game, including the starting state of the game. Complete information means that players possess all relevant information about their opponents, including their objectives, their possible actions, and their preferences. In economics, games are classified according to these two dimensions:
In the previous article, we talked about social deduction games — a genre of game where the influence currency (specifically, social influence) is used to mediate the initial imbalance between the power and information currencies of two teams. The information in social deduction games is imperfect and incomplete because while the rules, the objectives, and the possible information in the game are common knowledge, some players have no knowledge of the starting state of the game, or of which actions were taken by which players. Effectively, the “good” team in a social deduction game starts somewhere in the bottom-left quadrant and the “bad” team starts somewhere in the top-right quadrant; the bad team then tries to thwart the good team’s attempts to reach information parity.
In this article, we discuss perfect information games — the right half of our diagram. As it so happens, almost all of these games also have complete information. Perfect incomplete information games are (unsurprisingly) rare, but they do exist and I will touch on them in future articles.
Warming Up
Many beloved perfect information games started off as war games designed to help military generals sharpen their strategic thinking. Chess, xiangqi, and shogi likely have a common ancestor in chaturanga, a game played in the Gupta Empire in the 6th century AD. Chaturanga translates to “having four parts”, referring to the four divisions of an army — infantry, cavalry, chariotry, and elephantry — and involves encircling the enemy king (“checkmate” translates to “the king is dead” or “the king is helpless”, from the Persian shah mat). Go, also an encirclement game (but of territory, not of a particular piece), was created much earlier in China (possibly as early as 2000 BC) and eventually became xiangqi’s upper-class cousin, played by elites. These games have stood the test of time and are as much cultural artifacts as they are games. They have captivated generation after generation as players continue to uncover new insights about the games’ strategic depth — insights whose discovery is being accelerated today by artificial intelligence projects like AlphaGo. War games aside, though, “perfect information” is less a genre of game and more a specific property of the game’s information currency.
One consequence of having a perfect information game is that if the game mechanics allow players to directly interact, the game must be turn-based. If players were allowed to act simultaneously, then players would not have perfect knowledge of all previous actions in the game. Because perfect information games are often turn-based, the actions in these games (and the resulting game states) are amenable to being visually represented as trees. If you’ve ever taken an advanced course in game theory, probability, economics, or computer science, you’ll have seen diagrams like these:
We’ll avoid getting into the weeds of specific game trees in this article, but the one relevant concept here is the notion of a branching factor. For an individual node (game state) in the tree, the branching factor is the number of new nodes generated from the original one; each possible action you can take from that node gives rise to a new “branch” and a new resulting node. We are usually interested in the average or effective branching factor of a typical node at a given level of the tree.
The difficulty of a perfect information game is directly proportional to its average branching factor. Perfect information games can take years to master and are notoriously difficult — not because players lack meaningful information, but because they are overwhelmed by the burden of understanding its implications. You can only think so many moves ahead.
In the first article of this series, we discussed how currencies that never differ in quantity across teams can be excluded from our analysis. Since all players have access to exactly the same information in perfect information games by definition, information is not a relevant currency here. We are therefore left to grapple with just power and influence.
Power
Because we are by and large talking about turn-based games, one power currency that deserves attention is turn order.
In games with two teams, there is often a slight advantage associated with going first. This is especially true in games with high branching factors like chess and go, where small advantages accumulate like compound interest over several levels of the game tree to create winning states. Different games compensate for this known power imbalance in various ways — players may play multiple games as both sides in competitive settings, the advantaged side may be given a handicap, or the win condition for the advantaged side may be altered altogether. For example, in armageddon chess, the player with the White pieces is given more time on their clock, but Black wins the game if they secure at least a draw (while White has to win outright).
In games with three or more teams, the advantage of a particular position in the turn order becomes complicated to assess, especially if players can form impromptu, implicit agreements to eliminate other players first. Multiplayer variants of perfect information games have an even larger branching factor than their two-team counterparts, but they can be a great way to introduce a cooperative dynamic to games that makes turn order more irrelevant.
The most prominent power currency is the resources of the game itself — typically your pieces. In games like Ludo, Othello, and Go, all pieces move in the same way and therefore have equal value. Points are accumulated based on the particular arrangement of the pieces on the board (the theme generally being “capture territory”), not on the inherent value of the pieces themselves.
In games like checkers, chess, and shogi, pieces can move in different ways and therefore have different values. A numerical value might be assigned to the value of the pieces, but experts frequently disagree on what the correct values should be, especially if they are tangential to the game’s win condition (the theme generally being “capture your opponent’s most important piece”).
Most perfect information games start with equal resources for all teams. This simplifies the analysis of power considerably and makes the values of individual pieces less relevant because anything I can do to you, you can do to me. As the game progresses, the balance of power shifts, and in most situations, a decisive power advantage should lead to victory. However, power imbalances matter more in games which feature higher variance between the abilities of individual pieces. For example, in chess, the total power of your team is almost always decreasing, and the queen dominates every other piece by far. In such an ecosystem, it is almost impossible to recover from a significant power imbalance short of an immense amount of compensation in your influence currency. Contrast this with shogi, where pieces can reenter the game after they are captured, and the most powerful piece is only the rook. This often leads to mutual checkmating attacks even in the presence of significant power imbalance.
Nevertheless, different teams in perfect information games can start with different resources — or accumulate different resources along the way. Consider some variants of chess:
Or consider other genres of games altogether, like Bananagrams — a Scrabble-like game where you must use your own stash of lettered tiles to build a valid crossword before anyone else, while drawing additional tiles from a communal pile — and Monopoly. Because players can see everyone’s tiles in real time but can only interact with their own, Bananagrams is one of those rare perfect information games that is not turn-based. Players start off with different resources (i.e. different letters) and continue to receive different resources throughout the game, but the game is balanced around the fact that the English language contains enough words to accommodate most combinations of letters. Similarly, players in Monopoly accumulate different resources (money, properties, houses, hotels) along the way. The game is balanced around when these resources are dispensed, and what benefits are afforded to players who own a particular set of color groups, utilities, etc.
Just to drive the point home, these games are still perfect information games because all of these resources are known about from the beginning of the game. Players can plan around them — even when randomness plays a role in a player’s actions.
The last power currency worth mentioning is time. A player can spend endless amounts of time contemplating their move in a game with a high branching factor, so time is a very pragmatic currency for managing the duration of a game and encouraging more daring play. Typically time is implemented as a currency for the win condition: players lose if they run out of the time before their opponents (think chess clocks and increments, or the byoyomi system in go).
Influence
Unlike in social deduction games, where the influence currency is immediately accessible because of our innate familiarity with the “rules” of social interaction, influence in a perfect information game can be inscrutable to the novice. That is not to say that social influence is any less complex or multifaceted, but understanding influence in games where everything is known is a lifelong endeavor with many peaks to traverse and a near-infinite skill cap. To put this in perspective, consider two facts: 1) any decent AI for a perfect information game can easily defeat the human world champion in that game, and 2) the best AIs still lose to each other because they do not fully understand the influence currency.
At its core, your understanding of influence comes down to your ability to look at the current state of the game and assess how well each team is doing, what each team’s plan should be, and why. Perfect information games test how reliable your heuristics are for navigating and pruning the morass of possible actions you could take when you invariably find yourself in a part of the game tree that you’ve never encountered before — which is almost guaranteed to happen every time you play. Getting better at these games involves assessing myriad factors about a game state. Not just who has more resources (power imbalance), but who has more “space”, whose pieces are better “developed”, who has the “initiative”, who is “attacking” and “defending”, who is making better use of their “tempi”, and so on — where all these terms mean something specific in different games and are highly context dependent.
The critical skill in these games is knowing when and how to trade power for influence. This is something AI has been enlightening us on for a long time, especially so in recent years with the advent of neural-network AIs. These AIs do not rely on human-given heuristics. Instead, they build up their own positional understanding by playing games against themselves millions of times. I’ll discuss AlphaGo and AlphaZero in more detail in the next article.
Before next time…
In perfect information games, players are armed with the power currencies of turn order, resources, and time as they wade through a deluge of possible actions whose consequences cannot be calculated to endgame. Perfect information games compensate for the irrelevance of the information currency with a richness and inexhaustible depth in their influence currency.
With our introduction to social deduction games and perfect information out of the way, we have set the stage for a deeper exploration of the implications of the market economy framework. In the next article I will talk about the art of actually winning these kinds of games by carefully managing your power and influence currencies, which I will analogize to cash and unrealized gain in a stock market.
