Game Design and Mechanics
On Games, Part 4: Winning
In this article I make the case that the “power” and “influence” currencies introduced in the first article are analogous to the “cost” and “unrealized gain” of a portfolio in a market economy. I illustrate this with many games and argue that this analogy teaches us something about how to win these games.
So far we have talked abstractly about how power, influence, and information must be transferred and exchanged during the course of a game, through the game mechanics, to create a winning position for your team. In every example, influence functions as an intermediate currency where the game actually “happens” and the core skills of the game are tested. Ultimately, influence must be converted back into power because the win condition of a game is always tied to a power currency like time, points, a particular arrangement of pieces on a board, a particular sequences of events, etc.
In this article we are going to make this very explicit using analogies to actual markets, with the goal of identifying some common themes for winning games.
A Crash Course on Portfolios
A market is an organized mechanism that allows buyers and sellers to exchange goods, services, and information. An asset class is a group of financial instruments that behave similarly when traded in a market.
Some common asset classes:
- Cash
- Real estate — property, land, buildings
- Stock— an ownership stake in a company
- Bond — a loan made by an investor to a borrower
- Derivative — a contract between two parties that derives its value from an underlying asset
- Future — a derivative that obligates the parties to buy/sell an asset at a specific price on a specific date in the future
- Option — a derivative that gives buyers the right, but not the obligation, to buy/sell an asset at a specific price on a specific date in the future
Derivatives like futures and options have an expiration date. A derivative is said to be in the money if it will net a profit at the expiration date (assuming the current market price for the underlying asset holds), and out of the money if it will not.
A portfolio is a collection of investments in various asset classes. The cost, or book value, of a portfolio is the amount of cash that had to be spent to buy the initial set of assets. Assets change in value as they are traded on the market, resulting in a change to the market value of the portfolio.
The market value of a single asset is its quantity times its market price; the market value of the entire portfolio is the sum of the market values of the assets in the portfolio. The process of determining the market price of an assets through the interaction of buyers and sellers is called price discovery.
Unrealized gain is the potential profit resulting from an investment in an asset. Once an asset is liquidated (sold in the market), the actual profit resulting from the sale is the realized gain.
Trades often involve paying transaction fees. Realized gains are often taxed at a particular rate called the capital gains tax.
Mapping Games to Markets
Monopoly
Let’s think about Monopoly in terms of portfolios.
This is an easy one because Monopoly is precisely about accumulating and exchanging assets. You start the game with a portfolio of $1,500 in cash. Over the course of the game, your cash fluctuates as you accumulate other assets like income, properties (mortgaged or un-mortgaged), hotels, and houses. These assets are designed to function like real estate, or alternatively, like stocks that pay dividends (rent).
Your portfolio has a certain cost — this is the book value of all the assets you own (cash + how much you paid for all your properties + how much you paid for all your houses/hotels + price of un-mortgaged property + mortgage value of mortgaged property). Your portfolio also has an implicit market value — this includes the potential earnings you stand to make from other players paying you when they land on your properties. The market value of your portfolio doesn’t show up explicitly in the game — it is the hypothetical long-term value of your assets that accounts for all the different ways the game could play out. Buying houses and hotels increases the market value of your properties because you expect to generate returns on those assets later in the game.
The market value equals the cost plus some unrealized gain. Here’s the punchline: at any given time, the cost of your portfolio is precisely equivalent to your power currency, and the unrealized gain of your portfolio is precisely equal to your influence currency.
What insights about the game do we gain by representing Monopoly in this way? We observe that:
- Monopoly is essentially a game where you buy stocks and make moves in the game to increase the market value of those stocks (“buy low, sell high”).
- Monopoly is a perfect information game, hence the Monopoly market is “perfectly efficient”. This means that in theory, we could come up with accurate market values for different players’ portfolios that incorporated all available information. Given that every resource in Monopoly has a publicly known cost and income associated with it, estimating the market value of your portfolio is actually not difficult to do, especially in comparison to some other games we are about to discuss.
- In the Monopoly market, players have some degree of market control (i.e. they have some influence over the market values of their portfolios through their decision to buy/sell properties, build houses and hotels, etc.). However, dice rolls and chance cards introduce market shocks outside of the players’ control that cause market values to fluctuate.
Devising a “winning strategy” for Monopoly is therefore analogous to making investment decisions that will maximize the value of your portfolio:
- Determine the risk and return for each asset (every property, utility, house, hotel, etc.)
- Construct a portfolio that maximizes some utility function (e.g. mean variance utility)
- Dynamically rebalance your portfolio
This is precisely how tournament-level Monopoly players think about the game — their tips match up with a portfolio-driven approach to the game:
- Orange properties have the highest market value relative to their cost (one reason is because orange properties are an average dice roll away from the Jail square, which is the most frequented square on the board). The third house on a property creates disproportional returns —so get three houses on a property. Don’t buy hotels, which generate below-average returns because they put high-value houses back on the market.
- Buy every single property you land on — large market control and finite assets (there are only so many properties/houses) justify aggression as the path to maximizing utility.
- Mortgage order is a critical element of effectively rebalancing your portfolio, especially when you owe money. Monopoly quants have crunched the numbers and recommend mortgaging low-value properties, then high-value properties, then one utility, then one railroad, then a second utility.
In summary, framing Monopoly as a portfolio problem allows us to leverage tools from finance and portfolio theory when we think about how to win the game.
The act of translating each mechanic in a game to a finance concept might seem like overkill for a game like Monopoly, where players were likely thinking in those terms already. So let’s tackle a more non-trivial example. What other games function like stock markets?
Chess
Let’s think about chess in terms of portfolios.
Imagine a stock market where chess pieces are traded instead of companies. Instead of trading shares of Apple, Tesla, and Zoom, you can trade kings, queens, bishops, knights, rooks, and pawns. Each one of these assets has a cost in dollars, which we will approximate by the generally agreed-upon point values of pieces in chess:
When you start the game as either color, you are endowed with a portfolio of 8 pawns, 2 knights, 2 bishops, 2 rooks, 1 queen, 1 king, and $0 in imaginary cash. Nominally we will assign the king a cost of $1. The initial value of your entire portfolio is therefore $40. When one of your pieces is captured and comes off the board, the market (equivalently, the game) “buys back” your piece and gives you some imaginary cash in exchange. You lose the game if you are forced to sell your king back to the market (i.e. your king is liquidated).
Now, $40 is just the cost of your portfolio. Your portfolio’s market value changes throughout the game as you and your opponent start moving pieces. What determines the market value of a piece?
In real life, the market value of a stock like Apple is determined by a variety of factors. Market participants act on concrete financial information (e.g. its earnings last quarter), public announcements (e.g. Apple announces its next iPhone will come out in 9 months), and general optimism/pessimism about the company’s prospects. The market value of a piece in chess behaves similarly, driven by factors like how many squares it controls, how well it coordinates with other pieces, or how well it inhibits the movements of your opponent’s pieces — with subjectivity introduced into the calculation when you make a decision on how to measure those.
As soon as the game starts, most of your pieces are actually not worth their cost, because most of your pieces are not doing anything useful yet. It’s on you to get your pieces working. Furthermore, the values of all pieces in the game are highly correlated with each other — when you move one of your pieces, you affect the market value of your other pieces, and your opponent’s pieces, too. For example, the move e4 increases the e-pawn’s value, but it also increases the value of the queen and the light-squared bishop, which can now move:
Several moves in, the market value of your pieces will have fluctuated significantly the way actual stock prices do:
When a piece is eliminated from the board, it is liquidated at cost — even if the piece had a high market value during the game, the piece’s influence counts for nothing when the piece is removed from the board. Suppose your opponent captures one of your knights — the value of your portfolio goes down by the market value of the knight ($3 plus its unrealized gain before the capture), but the market only pays you $3 in cash for that knight. (It’s worth reiterating that you can’t do anything with this “cash”.) The unrealized gain is effectively a transaction cost that you incur when the knight leaves the board.
As in Monopoly, the market value of your portfolio is the sum of the market values of all your individual pieces. We can split that value into two components: the cost and the unrealized gain.
Again, we can ask ourselves — what insights about the game do we gain by representing chess in this way? We observe that:
- Like Monopoly, chess is a perfect information game where you start off with stocks and make moves in the game to increase the market value of those stocks. However, market value is much more difficult to estimate in chess because the game mechanics do not give you a precise way to quantify the unrealized gain.
- Unlike in the Monopoly market, players have complete market control — there are no market shocks or chance events.
- Unlike in Monopoly, assets can only be sold, not bought. New pieces are never placed on the board. (For sticklers: pawn promotion results in a large increase in the market value of that piece, but it is still the same piece.)
- Unlike in Monopoly, our portfolio value mostly decreases over time as assets (pieces) are liquidated (we are not counting our imaginary cash in the portfolio value). Power only increases via pawn promotion, which is a rare event.
Because there are no market shocks, there is no systemic risk. Because there is no element of chance in players’ moves (and hence the placement of pieces), there is no idiosyncratic risk either. Taken together, these differences mean that there is no risk in the game, and hence no point to diversification, rebalancing, etc. There isn’t even a mechanism to do this if you wanted to, because assets can only be sold.
When we consider these observations in aggregate, we arrive at some key insights:
- There is no point trying to increase the market value of one of your assets unless it is a) never liquidated, or b) liquidated in such that a way that your opponent’s portfolio loses at least as much market value as you do. Translation: don’t trade your best pieces, and capture your opponent’s best pieces.
- By the reverse logic, you should only liquidate an asset if your asset is unprofitable (market value < cost) and you can force your opponent to liquidate a profitable equal-cost asset (market value > cost). Translation: trade your worst pieces, and don’t capture your opponent’s worst pieces.
- You should also liquidate an asset if you believe you can make up the loss elsewhere by increasing the unrealized gain of another asset in your portfolio. Translation: gambits sacrifice power for influence. Successful gambits involve piece sacrifices that give rise to fierce attacks or overwhelming positional advantages several moves later.
- Your portfolio is more likely to have a higher market value than your opponent’s if you have a higher book value, more unrealized gain, or both. Translation: look for opportunities to reduce your opponent’s power (tactics — patterns of moves that win material or deliver checkmate) or influence (strategy — patterns of moves that create long-term positional advantages and turns pieces into “good” or “bad” pieces).
Chess engines actually give you all of this information. The “evaluation” of the position is a number that tracks which side is winning all throughout the game — this is exactly equal to the difference in portfolio values of White and Black.
Hopefully the usefulness of the portfolio analogy is becoming clear. While every game will have its own language for describing winning strategies and best practices, my claim is that these can be derived from first principles (or hinted at) by critically examining how the equivalent market for that game functions.
You can think of Shogi, Bughouse, and other chess-like games as more complicated stock markets where you can buy your opponent’s assets after they have been liquidated and add them to your portfolio, at the cost of a move. The “cash” in your portfolio can be interpreted as the number of moves you have left, and it now becomes relevant because it can be spent either on 1) moving a piece that’s already on the board (as in chess), or 2) buying one of your opponent’s liquidated pieces and placing it on the board. Players would therefore start with equally high (arbitrary) amounts of cash in their initial portfolio, and spend some cash for every move they made. The additional tradeoff of “do I make a move” vs. “do I place a piece” has to be factored into the market values of each asset, making them even harder to estimate.
I won’t belabor the translation of the next few games into portfolios as much, but let us briefly touch on games that involve trading derivatives, not stocks.
Go
In go, your pieces are valued not for their individual worth, but for the territory they cover as a group. Because you make moves with pieces but get points for intersections, go represents a different sort of portfolio from Monopoly and chess. Your portfolio in go comprises options, not stocks.
Intersections are the underlying asset, and each player is endowed with one option for every single intersection on the board. For argument’s sake, we can pretend that the White player has a $20 call option on every intersection (i.e. the right to “buy” the intersection at $20), and the Black player has a $10 put option on every intersection (i.e. the right to “sell” the intersection at $10). Every intersection starts with a market value of $15 (halfway between the strike prices). Players then make moves with their pieces, which causes the market values of the intersections to fluctuate. If the value of an intersection crosses $20, the White player can exercise their option on that intersection, which nets them a certain amount of cash; similarly, the Black player can exercise an option on a particular intersection if its value falls below $10. The winner is the player with the most cash at the end of the game, when all the profitable options have been exercised.
I won’t get into the nuts and bolts of go strategy here. I will simply highlight that many strategic concepts like ko fighting, thickness and lightness, sente and gote, etc. can be inferred by looking at how a seasoned trader might trade options for highly correlated assets (think of all the fancy-sounding options strategies that exist and how the composition of options in the portfolio reflects the investor’s beliefs about how the value of the underlying asset will change).
Games are often ended prematurely because both players agree which options will be “in the money” for them, and how much cash they would have ended up with, had the game been played to completion. Being able to tell with high accuracy which intersections you will capture is called “reading”. Intersections in go are informally classified as “alive”, “dead”, or “unsettled”. Equivalently: your option is in the money for you, in the money for your opponent, or out of the money for both players.
In real life, options are traded to hedge risk on the market price of the underlying asset in an intelligent way — the same is true in go. Because you have so many options in your portfolio (i.e. intersections where you can place stones), the game is a constant juggling act between making your own options profitable and preventing your opponent’s options from becoming profitable, and making progress on both tasks efficiently without wasting moves. Beginners often make the mistake of burning moves to capture territory too quickly and too early in places where they were likely to score points anyway.
Othello is another game that can be represented as a portfolio of options — you make moves with pieces, but get points for how many squares are flipped to your color by the end of the game. The underlying asset is therefore the squares, which fluctuate in market value as players make their moves.
Other Kinds of Portfolios
In roulette, you bet on a certain outcome by paying upfront with chips. Your bet is in the money if the outcome you predicted actually occurs. This is precisely how a future works. Your portfolio in roulette, then, consists of literal cash, and futures. You buy futures and immediately reap the profits (or losses) after each spin of the roulette wheel. Your power currency is your cash, and your influence currency is the unrealized gain from your futures. Alas, casino games like roulette are explicitly designed so that the expected value of those futures is less than the cost to purchase them. Moreover, in roulette, players have no market control.
Trick-taking card games like bridge, hearts, and spades also function as portfolios of futures, but with more market control. Your portfolio is your initial hand plus some imaginary cash (representing how many turns you have left); each card in your hand is effectively a future, and when you play a card, you are buying that future with some of your cash. The future is in the money when the card is the highest-value card that was played (per the rules of the specific card game), and if you win a trick, you are rewarded with points. Over many rounds, you are trying to accumulate as many points as possible. (You could also think of your hand as a portfolio of options, and you exercise one option every time it’s your turn.)
An imperfect, incomplete information game like Mafia functions a lot like the chess stock market — each player is like a chess piece, an asset with its own market value that gets liquidated when the player dies — but only certain dimensions of the overall portfolio values are known. As a Villager, you might know exactly how many Mafia and how many Villagers are left (power), and maybe you can intuit which players are exerting the most control over the game during the day phase (influence), but because you do not know which players belong to which teams, you cannot accurately sum these to estimate the portfolio value of both teams (power + influence). An important difference from chess is that when you are eliminated in Mafia, you incur a smaller transaction cost: some of your influence in the game survives you, because people can remember what you did and said. The information about team identity is precisely what you are trying to infer from the trajectory of the assets whose market values you can estimate.
The Sticking Point: Estimating Market Value
Winning a game involves two skills. The first skill is evaluation, or understanding your current standing — how close are you to winning and why? The second skill is decision, or picking the right choice among myriad choices — which action will maximize my chance of winning in the future?
Any value to be derived by borrowing concepts from finance to further our understanding of game dynamics hinges on players being able to assess the market value of everyone’s assets with some degree of accuracy. The more accurately you can evaluate a portfolio, specifically the value of influence (unrealized gain), the better your chance of making progress towards the goal of winning.
In reality, evaluation is incredibly difficult. It can take years to look at any of the games I’ve mentioned in these articles and fully internalize everything that is happening in the game at any given time. In perfect information games, computers have long surpassed humans, and we celebrate those rare moments when humans have actually outwitted their AI counterparts. Think Lee Sedol’s inspired move against AlphaGo in Game 4 of their epic 2016 encounter, or checkers champion Marion Tinsley’s logic-defying “you’re gonna regret that” comment to Chinook, the best checkers AI in 1990, after Tinsley looked a dizzying 64 moves ahead to divine that he would beat the computer (and did). (Checkers was solved to a draw in 2007.)
In imperfect information games too, like poker, modern AIs like Pluribus can estimate probabilities better than any human, and play nuanced mixed strategies with flawless execution. Yet even our most cutting-edge game AIs disagree on evaluation. In the 13th Computer Chess Championship from this year, the top two chess engines — Leela Chess Zero and Stockfish — frequently disagreed on evaluation, with commentators identifying several positions in which both engines thought they were definitively winning.
Evaluation is difficult because influence is an irreducibly complex currency. AI is helping us plumb the depths of just how much influence can compensate for a lack of power. Today’s neural-net chess AIs, perfected by playing millions of games against themselves without human intervention of any kind, prefer initiative over materialism, and frequently give up power to obtain an influence advantage that only becomes apparent several moves later. Poker AIs can routinely play mediocre hands successfully because of their immense insight into effective bidding (low power vs. high influence). Tying this back to the theme of this article, AI in games promises to be the platonic ideal of a stock-picker — willing to splurge its resources on investments in a way that humans wouldn’t because it can recognize long-term unrealized gain.
Before next time…
To master a game is to learn how to improve the quality of our resources over time, and stymie our opponents in their efforts to do the same. Winning is a dance between power and influence that mirrors the dance between book value and unrealized gain in a portfolio; analogizing games to trading stocks, options, and futures can help us understand what kind of game we are playing.
