# Game Theory: Applications in Trading & Investing

Game theory is a broad field that studies how people make decisions in strategic situations.

We’ll define game theory concepts in the first section of this article.

In the following section, we’ll look at these concepts’ potential application to finance, trading, and investing.

## Key Takeaways – Game Theory: Applications in Trading & Investing

Game Theory’s Scope: Game theory explores decision-making in strategic scenarios, encompassing a wide array of principles and strategies.Applying Game Theory Concepts to Finance: Game theory concepts find practical use in finance, trading, and investing by optimizing portfolios, understanding market dynamics, and analyzing competitive interactions.

Below we have a list of some fundamental principles and concepts in game theory:

## Game Theory Concepts

### Expected Utility (Expected Value)

It reflects the average payoff of a decision when factoring in all possible outcomes and their associated probabilities.

Players aim to maximize their expected utility.

### Strategic Dominance

**Strict Dominance**: A strategy is strictly dominant if it always yields a higher payoff than another strategy, no matter what the opponents do.**Weak Dominance**: A strategy is weakly dominant if it yields at least as high a payoff as any other strategy and there’s at least one scenario where it does better.

### Nash Equilibrium

A situation in which each player’s strategy is optimal given the strategies chosen by the others.

No player has an incentive to deviate unilaterally from their current strategy.

### Mixed Strategy Nash Equilibrium

A situation in which players randomize over multiple strategies in a way that makes no player want to deviate.

### Zero-Sum Games

Games in which one player’s gain is exactly balanced by the other player’s loss.

### Cooperative vs. Non-Cooperative Games

Cooperative games study how players can collaborate and form coalitions, while non-cooperative games focus on individual decision-making.

### Subgame Perfect Equilibrium

A refinement of Nash Equilibrium, it ensures that players’ strategies are optimal even in every subgame (or portion) of the larger game.

### Backward Induction

A method used to solve finite extensive-form or sequential games.

It involves working backward from the end of the game to determine the optimal strategy at every stage.

### Payoff Matrix

A table that lists out the payoffs for each combination of players’ strategies.

### Pure vs. Mixed Strategies

A pure strategy involves always choosing a specific action, while a mixed strategy involves randomizing over several actions.

### Prisoner’s Dilemma

A canonical game illustrating the tension between individual and group rationality.

Two players can choose to cooperate or betray the other.

Individual rationality leads to betrayal, but both players are better off if they cooperate.

### Stag Hunt

Another classical game that represents a situation with two Nash Equilibria, one risk-dominant and the other payoff-dominant.

### Pareto Efficiency

A situation is Pareto efficient if there’s no other feasible outcome that would make at least one player better off without making anyone else worse off.

### Common Knowledge

Information that everyone knows and everyone knows that everyone knows (and so on).

### Evolutionary Stable Strategy (ESS)

A strategy that, if adopted by a population in a game, cannot be invaded by any alternative strategy that is initially rare.

### Repeated Games

Games that are played multiple times by the same players.

This repetition can lead to different strategies and outcomes compared to one-shot games.

### Minimax Theorem

In zero-sum games, a player will choose a strategy that maximizes their minimum payoff, ensuring the best worst-case scenario.

### Kuhn’s Theorem

In poker-style games, if a strategy is part of a mixed-strategy Nash equilibrium, then every hand that is played with positive probability in that strategy must yield the same expected value.

### Correlated Equilibrium

A generalization of Nash equilibrium where players condition their strategies on signals from an external random device.

### Bayesian Games

Games in which players have incomplete information about the other players, like their payoffs or strategies, and have to make decisions based on probabilities.

### Mechanism Design

The “inverse” of traditional game theory.

Instead of analyzing given games and finding equilibria, it’s about designing a game (or mechanism) to achieve a particular outcome.

### Symmetric vs Asymmetric Games

**Symmetric**: Players have the same set of strategies and payoffs. The game looks the same from each player’s perspective.**Asymmetric**: Players may have different strategies, payoffs, or information.

### Zero-Sum vs Non-Zero-Sum Games

**Zero-Sum**: The total gains and losses among participants are always equal, balancing out to zero. One player’s gain is precisely another’s loss.**Non-Zero-Sum**: The total gains and losses can be greater or less than zero. Collaboration or competition can result in outcomes where everyone benefits or everyone loses.

### Simultaneous vs Sequential Games

**Simultaneous**: Players make decisions at the same time without knowledge of the other player’s decision.**Sequential**: Players take turns making decisions, often with knowledge of previous actions.

### Perfect Information vs Imperfect Information

**Perfect Information**: All players have complete knowledge about all elements of the game, including past moves.**Imperfect Information**: Players don’t have complete knowledge of all the elements. This could mean they don’t know previous moves, other player’s payoffs, etc.

### Bayesian Game

A game in which players have incomplete information about other aspects of the game, typically the other player’s type or payoffs.

Players have beliefs about these types, represented by probability distributions.

### Combinatorial Games

Games with no randomness or hidden information.

Players take turns making moves, and the outcome (win, lose, or draw) is determined by the combination of moves made by all players.

### Infinitely Long Games

These games do not necessarily end after a fixed number of moves.

They often require a different set of tools and techniques, like the use of transfinite numbers.

### Discrete vs Continuous Games

**Discrete**: Players choose from a finite number of strategies.**Continuous**: Strategies are chosen from a continuous set, often represented by a range of numbers or actions.

### Differential Games

A type of dynamic game where the evolution of state variables is governed by differential equations.

Often used in fields like economics and control theory.

### Evolutionary Game Theory

Studies the behavior of large populations of agents over time, focusing on how strategies evolve.

Strategies that yield higher payoffs tend to become more common over time.

### Stochastic Outcomes

Games in which outcomes are determined not just by the strategies of the players but also by random events.

Players often have to optimize expected payoffs.

### Metagames

Games about games.

This could involve creating strategies for a base game or even considerations about which game to play.

### Pooling Games

A type of game where players can form coalitions and pool resources to obtain some advantage.

The challenge is often how to distribute the gains from cooperation among the coalition members.

### Mean Field Game Theory

Studies games with a large number of players where each player’s impact is negligible.

Players react to the “average effect” of the population, which is where the term “mean field” comes from.

## Game Theory Principles – Applications in Finance, Trading, and Investing (Examples)

Let’s break these down and explore how they might be applied in the context of finance, trading, and investing:

### Expected Utility (Expected Value)

Portfolio optimization is about maximizing expected utility.

A trader/investor might weigh the expected returns of various assets against their associated risks, determining an optimal asset allocation that best suits their risk tolerance and investment horizon.

### Strategic Dominance

In algorithmic trading, certain trading strategies might be identified that consistently outperform other strategies, regardless of market conditions (strict dominance).

Alternatively, strategies might be identified that outperform in specific conditions and never do worse than break even in other conditions (weak dominance).

### Nash Equilibrium

Consider two firms deciding whether to enter a market.

If both enter, they might split the profits, but if one stays out, the other could capture a larger share.

The strategies the firms adopt, knowing the strategies of the other firm, could settle into a Nash equilibrium.

### Mixed Strategy Nash Equilibrium

Traders might randomize their order placements (between market orders, limit orders, and so on) to prevent predictability and exploit market conditions.

### Zero-Sum Games

Alpha generation in markets is a zero-sum game, as one person’s gain comes at the expense of another’s loss.

### Cooperative vs. Non-Cooperative Games

Firms might collaborate to set prices or form cartels (cooperative games).

In contrast, competitive pricing models, where firms set prices independently, represent non-cooperative games.

### Subgame Perfect Equilibrium

Multi-stage M&A negotiations can be viewed through this lens, ensuring that strategies are optimal at each stage of the negotiation.

### Backward Induction

Used in figuring out what one’s goals are and then working backward to figure out the specific steps needed to achieve it.

For example, say someone was making a hefty time commitment to starting a successful YouTube channel and wanted to earn $10,000 per month in ad revenue.

Let’s say they earned a $6 RPM ($6 per 1,000 views) and each video achieved an average of 1,000 views per month.

They could then calculate how many videos they would need in order to achieve this goal:

Videos needed = $10,000 / ($0.006 * 1,000) = 1,667 videos

Of course, there are other steps that need to be taken as well (e.g., what type of videos to make, scripts, editing, etc.) to achieve the desired result.

### Payoff Matrix

In trading, the payoff matrix might be used to determine the profits/losses based on different scenarios playing out.

### Pure vs. Mixed Strategies

A trader might always use a certain strategy when certain market conditions are met (pure strategy) or use a probabilistic mix of strategies (mixed strategy).

### Prisoner’s Dilemma

Two firms might benefit from cooperation, such as by both keeping prices high to maximize profits.

However, the temptation for one firm to undercut the other’s price to capture more market share can lead to both firms reducing prices, earning less profit – a scenario reminiscent of the Prisoner’s Dilemma.

### Stag Hunt

Two investment firms might choose between investing in a stable, known commodity (like short-term government bonds) or a new, riskier venture.

If both firms invest in the new venture, it could succeed and yield high returns.

But if only one firm invests, the venture might fail due to a lack of support.

### Pareto Efficiency

Portfolio optimization seeks allocations that are Pareto efficient, where no other allocation improves the return for one asset without decreasing the return for another.

### Common Knowledge

Publicly available financial reports create a common knowledge base among traders.

Everyone knows the information, and everyone knows that everyone else knows it.

Everything that’s known is generally already discounted in the price.

### Evolutionary Stable Strategy (ESS)

Over time, trading strategies that yield consistent profits might become more prevalent, while less profitable strategies become rare or extinct.

### Repeated Games

Ongoing relationships between banks and borrowers can be viewed as repeated games.

Trust and reputation become critical, and strategies might differ from one-off transactions.

### Minimax Theorem

In highly speculative trading environments, traders might employ strategies that aim to maximize their potential gains while minimizing potential losses, especially in zero-sum situations.

For example, they might balance their portfolio better or use options to eliminate left-tail risk.

### Kuhn’s Theorem

Much like in poker, traders using mixed strategies might ensure that various “plays” or trades, when played with positive probability, have the same expected value to remain unpredictable.

### Correlated Equilibrium

Traders might base their strategies on signals from external indicators or benchmarks.

### Symmetric vs. Asymmetric Games

**Symmetric**: Consider two hedge funds with identical strategies. (For example, a long-time lieutenant of an investor goes off to start their own firm.) Their competition might reflect a symmetric game since both have the same training and knowledge at their disposal.**Asymmetric**: Retail investors vs. institutional investors. Institutional investors often have access to better information, tools, and financial instruments, making the investment game asymmetric. Can also relate to company management knowing more than external investors.

### Zero-Sum vs. Non-Zero-Sum Games

**Zero-Sum**: Tactically trading the markets can be viewed as zero-sum as one person’s gains are another’s losses.**Non-Zero-Sum**: Investing (beta) is typically non-zero-sum. All investors can see gains if the market goes up, or all can experience losses if the market drops. Productivity that leads to rising living standards is non-zero-sum.

### Simultaneous vs. Sequential Games

**Simultaneous**: IPO pricing can be viewed as a simultaneous game where investment banks and initial investors decide on the purchasing price without knowing the other’s valuation.**Sequential**: Consider a scenario where one company is considering a takeover of another. The target company makes a move after observing the actions of the acquiring company.

### Perfect Information vs. Imperfect Information

**Perfect Information**: Chess can be a metaphor here – like an open market where all moves (transactions) are publicly available.**Imperfect Information**: Most real-world investing scenarios operate under imperfect information, where investors don’t have access to all information or future events impacting asset values. Poker is another game example. Imperfect information also tends to create more variance in the outcomes.

### Bayesian Games

Investment decisions often have to be made with incomplete information.

Investors might use Bayesian updating to adjust their beliefs based on new information, such as earnings reports.

### Combinatorial Games

Complex financial instruments with multiple options (e.g., exotic options) can be seen as combinatorial games where different combinations of decisions lead to different outcomes.

As another example, complex financial deals, especially those involving multiple stages or clauses (like staged investments in venture capital), can be likened to combinatorial games where each combination of moves (decisions) yields different outcomes.

### Infinitely Long Games

Some investments, like endowments or trusts, can have horizons that are effectively infinite.

The strategies for such investments would be very different from those with shorter time horizons.

For example, they might include more illiquid investments and more strategic and less tactical approaches to capital allocation.

### Discrete vs. Continuous Games

High-frequency trading (continuous) might involve making decisions at every tiny fraction of a second, whereas some investment decisions might be made at discrete intervals (e.g., quarterly).

### Differential Games

In dynamic portfolio management, where assets’ performances are governed by differential equations (like the Black-Scholes model for option pricing), differential games provide insight into optimal strategies.

### Evolutionary Game Theory

Investment strategies might evolve over time based on their success.

Successful strategies are more likely to be replicated and adapted by others.

This forces additional evolution because an edge in the market will be eroded if it’s copied at a great enough scale.

### Stochastic Outcomes

Many financial models incorporate random variables to account for uncertainty, such as in the pricing of derivatives.

### Metagames

A metagame in the context of financial markets refers to the overarching strategies and considerations that traders and investors use, which go beyond the basic analysis of individual assets.

Deciding which specific markets or assets to trade in (e.g., stocks, commodities, foreign exchange) and how to combine returns streams in an efficient way based on meta-strategic considerations can be seen as a metagame.

Since the metagame concept is important in trading, let’s dedicate a bit more space to it.

Some key points to understand about the metagame in financial markets:

#### Behavioral Analysis

The metagame often involves understanding the psychology and behavior of other traders and investors.

For example, if a trader believes that most other traders are bullish on a particular stock and overweighted, they might decide to go against the grain and short the stock, anticipating a future price drop.

#### Macro Considerations

While individual asset analysis (like stock analysis) looks at the specifics of a company or asset, the metagame might involve considering broader economic, political, or global factors that could influence the market as a whole.

They might consider how portfolio construction with a broader set of assets can enhance returns and reduce risk versus being concentrated in a particular asset class, even if they happen to be a specialist in it.

#### Anticipating Trends

Part of the metagame is trying to anticipate shifts in market sentiment or trends.

This could involve recognizing early signs of a market bubble or identifying potential shifts in monetary policy that could affect asset prices.

#### Strategic Positioning

Traders and investors might position their portfolios based on their understanding of the metagame.

For instance, if they believe that alpha is not easy to achieve in most markets, they might learn how to balance their portfolio to enhance the return they get per each unit of risk.

#### Information Asymmetry

The metagame can also involve leveraging information that might not be widely available or understood by the broader market.

This doesn’t mean insider trading (which is illegal), but rather understanding nuances or implications of available information that others might overlook.

For example, a trader with a legal background is more likely to understand the nuances of certain lawsuits are likely to have on a company.

#### Feedback Loops

In financial markets, the actions of traders and investors can create feedback loops.

For example, if many traders believe a stock will go up and start buying, their actions can actually drive the price up. Understanding these feedback loops is a crucial part of the metagame.

#### Overall

In essence, the metagame in financial markets is about understanding the “game” that everyone else is playing and making strategic decisions based on that understanding.

It’s a higher-level strategy that takes into account the actions, beliefs, and strategies of other market participants.

### Pooling Games

Investment pools or syndicates, where multiple investors pool their resources to invest in larger projects, distribute the returns based on the input and agreed-upon rules.

Syndication is common for large private equity deals. It helps to prevent one project from taking up too large a fraction of their portfolio.

### Mean Field Game Theory

In highly liquid markets, an individual trader’s actions might have a negligible effect, but the aggregate behavior of all traders determines market dynamics.

## Definitions of Game Theory Terms

### Congestion Game

A game in which players choose a subset of available resources, and the payoff for each player depends on the number of players choosing the same resources. The more congested a resource, the lower the payoff.

### Cooperative Game

A game where players can form binding commitments or coalitions with one another to achieve mutual benefits, rather than competing individually.

### Determinacy

A property of a game indicating that there exists a strategy for one player that guarantees a certain outcome, regardless of the strategies chosen by other players.

### Escalation of Commitment

The tendency for players to continue investing in a decision based on the cumulative prior investment, despite new evidence suggesting that the decision was probably wrong.

### Extensive-form Game

A representation of a game that specifies the order in which players make decisions, possible actions at each decision point, and payoffs for all possible combinations of actions.

### First-player and Second-player Win

Terms used in combinatorial game theory to denote games where either the first player (with optimal play) is guaranteed a win or the second player is.

### Game Complexity

A measure of the computational difficulty of finding an optimal strategy or determining the outcome of a game.

### Graphical Game

A game in which the interactions between players are represented by a graph, where each node represents a player and edges represent direct interactions between players.

### Hierarchy of Beliefs

A model that captures players’ beliefs about the actions of other players, their beliefs about others’ beliefs, and so on.

It’s a bit of a “they think that I think that they think…” type of concept.

### Information Set

A set of decision points in a game where a player cannot distinguish between the different points based on the moves previously made.

### Normal-form Game

A representation of a game in matrix form, where players’ strategies and the corresponding payoffs are tabulated.

### Preference

A player’s ranking or valuation of possible outcomes in a game, indicating which outcomes they like more or less relative to others.

### Sequential Game

A game where players make decisions one after another, rather than all at once.

### Simultaneous Game

A game where all players make decisions at the same time, without knowledge of the choices being made by their opponents.

### Simultaneous Action Selection

A scenario in which all players in a game choose their actions at the same time, without observing the choices of others.

### Solved Game

A game for which the optimal strategies for all players have been determined, and the outcome is known given those strategies.

### Succinct Game

A game that can be described in a concise manner, often using a compact representation, but determining an optimal strategy might be computationally difficult.

## Equilibrium Concepts

### Bayesian Nash Equilibrium

A solution concept where each player’s strategy maximizes their expected payoff, given their beliefs about the other players’ strategies and types.

### Berge Equilibrium

A solution concept where each player’s strategy is a best response to a set of strategies that are beneficial for the other players.

### Core

In cooperative game theory, the set of payoff allocations where no group of players can achieve a better outcome by deviating and cooperating among themselves.

### Correlated Equilibrium

A solution concept where players choose their strategies based on a correlated random device, and no player has an incentive to deviate from the recommended strategy given the strategies of others.

### Epsilon-equilibrium

A relaxation of Nash equilibrium where each player’s strategy is nearly optimal, allowing for a small deviation of ε.

### Evolutionarily Stable Strategy (ESS)

A strategy that, when adopted by a population, cannot be invaded by any alternative strategy due to its higher fitness.

### Gibbs Equilibrium

A generalization of Nash equilibrium in games with continuous strategies, based on the principles of statistical mechanics.

### Mertens-stable Equilibrium

A refinement of Nash equilibrium that remains stable even when players consider the possibility of future deviations.

### Markov Perfect Equilibrium

A solution concept for dynamic games where players’ strategies are based solely on the current state of the game and are optimal given the strategies of other players.

### Nash Equilibrium

A situation in which each player’s strategy is optimal given the strategies chosen by all other players.

### Pareto Efficiency

An allocation of outcomes where no player can be made better off without making at least one other player worse off.

### Perfect Bayesian Equilibrium

A refinement of Bayesian Nash equilibrium in dynamic games where players update their beliefs consistently using Bayes’ rule.

### Proper Equilibrium

A refinement of Nash equilibrium that eliminates weakly dominated strategies and is robust to small perturbations in the game.

### Quantal Response Equilibrium

A solution concept where players’ strategies are probabilistic responses to the expected payoffs from other players’ strategies.

### Quasi-perfect Equilibrium

A refinement of subgame perfect equilibrium that accounts for off-the-equilibrium-path beliefs.

### Risk Dominance

A criterion used to select among multiple Nash equilibria based on their riskiness.

### Satisfaction Equilibrium

A situation where each player’s strategy meets a personal satisfaction level, rather than necessarily maximizing payoff.

### Self-confirming Equilibrium

A solution concept where players’ strategies are optimal given their beliefs, even if those beliefs are incorrect but are never contradicted by observed play.

### Sequential Equilibrium

A refinement of Nash equilibrium for dynamic games that considers consistency of beliefs and strategies throughout the game.

### Shapley Value

In cooperative game theory, a method to fairly allocate the total value of a game among players based on their marginal contributions.

### Strong Nash Equilibrium

A Nash equilibrium that remains stable even when any coalition of players deviates jointly.

### Subgame Perfection

A refinement of Nash equilibrium that requires strategies to be optimal in every subgame of the original game.

### Trembling Hand

A concept used to refine equilibria by considering games where players may make small mistakes with a tiny probability.

## Strategies in Game Theory

### Backward Induction

Backward induction is a method used in dynamic games where players deduce the optimal actions by analyzing the game from its end to its beginning.

By considering the final stages first, players can determine the best moves at earlier stages.

### Bid Shading

Bid shading is the practice of bidding less than one’s true valuation in an auction, often to account for the winner’s curse or to avoid overpaying.

### Collusion

Collusion occurs when players in a game coordinate their actions to achieve a mutually beneficial outcome, often at the expense of other players or external parties.

This coordination is typically against the rules or laws governing the game.

### Forward Induction

Forward induction is a reasoning method in dynamic games where players deduce the optimal actions by analyzing the game from its beginning to its end.

Players infer past actions and beliefs based on observed strategies and then determine future actions accordingly.

Chess would be a popular example of a pure forward induction game (i.e., backward induction in chess doesn’t make sense).

### Grim Trigger

Grim trigger is a strategy in repeated games where a player cooperates until the opponent defects.

Once the opponent defects, the player retaliates by defecting indefinitely, “pulling the trigger” on the cooperative relationship.

### Markov Strategy

A Markov strategy is a strategy in a dynamic game where a player’s action depends only on the current state of the game and not on past history.

The strategy is memoryless and is based on the current observable variables.

In portfolio theory, one might think of it as assessing the portfolio as-is and not being biased to hold a position just because it’s there.

### Dominant Strategies

A dominant strategy is a strategy that yields the highest payoff for a player regardless of the strategies chosen by other players.

If a player has a dominant strategy, they will always play it.

### Pure Strategy

A pure strategy is a game strategy where a player takes a specific action with certainty.

There’s no randomness or probabilistic element involved in the decision.

### Mixed Strategy

A mixed strategy involves a player randomizing over multiple pure strategies, choosing each with a specific probability.

It introduces an element of unpredictability to a player’s actions.

Poker is an example of a mixed-strategy game. Playing the same strategy all the time can lead to predictability and thus exploitability by other players.

For example, in certain situations, a player might bet 1.5x pot 70% of the time and bet 0.6x pot 30% of the time, or call 80% of the time in a certain situation but raise 20% of the time.

### Strategy-stealing Argument

The strategy-stealing argument is a method used in combinatorial games to prove that the first player has a winning strategy.

It assumes that the second player has a winning strategy and then demonstrates a contradiction by showing that the first player can “steal” or adopt a version of this strategy to win.

### Tit for Tat

Tit for tat is a strategy in repeated games where a player starts by cooperating and then mimics the opponent’s previous action in subsequent rounds.

If the opponent cooperated in the last round, the player cooperates in the next; if the opponent defected, the player defects in the next round.

## Classes of Games

### Bargaining Problem

A bargaining problem refers to a situation where two or more players negotiate over the division of a surplus.

The goal is to reach an agreement that is beneficial for all parties involved, taking into account each player’s preferences and threats.

### Cheap Talk

Cheap talk refers to communication between players that does not directly affect the payoffs of the game but can influence players’ beliefs and actions.

The communication is non-binding and costless.

### Global Game

A global game is a game where players receive slightly noisy private information about an underlying state of the world and then choose actions based on this information.

It helps in analyzing situations where there is strategic uncertainty.

### Intransitive Game

In an intransitive game, player preferences are not transitive, meaning that there isn’t a clear hierarchy of strategies.

For example, strategy A might beat strategy B, B might beat C, but C might beat A.

For example, this might occur in sports where A is expected to beat B, and B is expected to beat C, but C might be expected to beat A.

This might generally have to do with relative strengths and weaknesses and transitivity logic might not apply.

### Mean-field Game

A mean-field game involves a large number of players where each player’s payoff depends on their own action and the average of the actions of all other players.

It captures the idea that each individual’s impact on the overall system is negligible.

Voting might be an example.

The odds of someone’s individual vote in the US presidential election affecting the result is effectively zero.

### Mechanism Design

Mechanism design is the “reverse” of traditional game theory.

Instead of starting with a game and analyzing strategies, mechanism design begins with a desired outcome and designs a game or mechanism to achieve that outcome.

### n-player Game

An n-player game is a game with ‘n’ distinct players. It generalizes games beyond the typical two-player setup.

### Perfect Information

A game of perfect information is one where all players are fully aware of all the actions that have taken place previously.

Chess is a classic example.

### Large Poisson Game

In a large Poisson game, the number of players is random and follows a Poisson distribution.

Each player decides on an action without knowing the exact number of participants.

Trading might be an example.

You might make a trade in the markets without knowing exactly how many participants there are (or how big they are and what their motivations might be).

### Potential Game

A potential game is a game where any unilateral change by a single player leads to a corresponding change in a potential function.

This potential function helps in analyzing and predicting the game’s equilibria.

In a specific financial market, the actions of one player might have impacts on other players as well.

### Repeated Game

A repeated game is one where the same game (or stage game) is played multiple times in succession.

The history of past plays can influence strategy choices in future rounds.

### Screening Game

A screening game involves an uninformed player taking an action to induce the informed player to reveal private information.

It’s a type of asymmetric information game.

### Signaling Game

In a signaling game, an informed player takes an action to convey private information to an uninformed player.

The uninformed player then takes an action based on the received signal.

### Stackelberg Competition

In Stackelberg competition, one player (the leader) moves first and the other player (the follower) moves after observing the leader’s action.

It’s a model used in oligopoly theory.

### Strictly Determined Game

A strictly determined game is a two-player zero-sum game where the value of the game is the same regardless of the strategy chosen by the player who moves first. Chess would be an example.

### Stochastic Game

A stochastic game is a dynamic game with probabilistic transitions. Players’ actions not only determine payoffs but also influence the probability distribution over future states.

### Symmetric Game

In a symmetric game, all players are identical in terms of strategy choices and payoffs.

If any player swaps places with another, the game’s outcome remains unchanged.

### Zero-sum Game

A zero-sum game is a game where the total gain (or loss) of utility is zero when considering all players.

The gain (or benefit) of one player is exactly balanced by the losses of other players.

Wins and losses in sports would be an example.

## Popular Games in Game Theory & Relations to Finance and Trading

Let’s look into each game and then explore their relation to finance, trading, and investing:

### Go

An ancient board game where two players use black and white stones to capture territory. The objective is to control a larger area than the opponent.

**Relation to Finance**: Go emphasizes long-term strategy and adaptability, much like trading strategies where one must anticipate market movements and adjust portfolios accordingly.

### Chess

A two-player strategy board game where each player controls an army with different types of pieces, aiming to checkmate the opponent’s king.

**Relation to Finance**: Chess requires strategic foresight and planning several moves ahead, similar to financial planning and forecasting in trading and investing.

### Infinite Chess

A variant of chess played on an unbounded chessboard.

**Relation to Finance**: Represents the idea of infinite possibilities and outcomes, akin to the “open system” nature of financial markets (where there are lots of known unknowns and unknown unknowns) relative the “closed system” of traditional chess.

### Checkers

A two-player game where players move their pieces diagonally and capture opponent’s pieces by jumping over them.

**Relation to Finance**: Checkers, like trading/investing, requires recognizing patterns and making strategic decisions based on limited moves.

### Tic-tac-toe

A simple game where two players take turns marking spaces in a 3×3 grid, aiming to get three of their marks in a row.

**Relation to Finance**: You should never lose at tic-tac-toe as long as you know to occupy the middle square. Just as you should never lose all your money in trading as long as you diversify and use options to eliminate your left-tail risk.

### Prisoner’s Dilemma

A standard example of a game that shows why two rational individuals might not cooperate, even if it appears that it’s in their best interests to do so.

**Relation to Finance**: Highlights the challenges of cooperation in financial markets, such as cartel agreements in price setting or the dilemma faced by traders in collusive practices. Can also relate to how countries might behave with other in terms of geopolitical risks to portfolios.

### Gift-exchange Game

A game that examines reciprocity where one player (the employer) offers a wage, and the other (the worker) responds with an effort level.

**Relation to Finance**: Reflects employer-employee relationships in finance, emphasizing the importance of incentives and trust.

### Optional Prisoner’s Dilemma

A variant of the Prisoner’s Dilemma where players can opt out of playing the game.

**Relation to Finance**: Demonstrates the choice investors/traders might have to participate (or not) in certain market conditions or investment opportunities.

### Traveler’s Dilemma

A non-zero-sum game where two players price an identical item; the lower price is rewarded, but the player who quotes the lower price gets a smaller bonus.

**Relation to Finance**: Reflects pricing strategies in competitive markets and the balance between undercutting competitors and maximizing profit.

### Coordination Game

A game where players benefit from coordinating their actions but have multiple equilibria to choose from.

**Relation to Finance**: Represents scenarios where multiple financial strategies can be optimal, and market participants need to anticipate others’ actions.

### Chicken

A game where two players drive toward each other, and the one who swerves is the “chicken”. Both swerving or not swerving have their payoffs.

**Relation to Finance**: Analogous to trading in a bubble environment where traders decide whether to hold their position or exit, knowing the price run-up isn’t sustainable, anticipating the moves of others.

### Centipede Game

A game of multiple rounds where players decide whether to take a larger share of an increasing pot or pass it to the other player.

**Relation to Finance**: Reflects situations where traders must decide between taking profits now or waiting for potentially larger gains, risking loss.

### Lewis Signaling Game

A game where one player sends a signal, and the other responds. The game examines how communication can lead to coordination.

**Relation to Finance**: Relevant in scenarios where market signals (e.g., interest rate changes) prompt responses from traders and investors.

### Volunteer’s Dilemma

A game where one player’s volunteering provides a public good, but at a cost to the volunteer.

**Relation to Finance**: Reflects situations where one party bears a cost for a collective benefit, like a company investing in public infrastructure. In this case, the payoff may be far in the future (or never).

### Dollar Auction

An auction where players bid on a dollar, but both the highest and second-highest bidders pay, leading to bids exceeding one dollar.

**Relation to Finance**: Demonstrates the irrational escalation in financial markets, where participants might overpay due to competition.

### Battle of the Sexes

A coordination game where two players prefer different outcomes but want to coordinate their actions.

**Relation to Finance**: Represents scenarios where market participants have different preferences but benefit from market stability and predictability.

### Stag Hunt

A game where two players can hunt a stag (cooperating) or a rabbit (not cooperating). Hunting a stag provides more meat but requires cooperation.

**Relation to Finance**: Highlights the benefits of collaboration in financial ventures, where joint efforts can lead to larger rewards.

### Matching Pennies

A two-player game where players reveal a penny heads up or tails up. One player wins if the pennies match, while the other wins if they don’t.

**Relation to Finance**: Represents zero-sum situations in trading where one’s gain is another’s loss.

### Ultimatum Game

One player proposes a division of a sum of money, and the other can accept or reject. If rejected, neither gets anything.

**Relation to Finance**: Reflects power dynamics in financial negotiations, emphasizing fairness and rationality.

### Rock Paper Scissors

A hand game where each shape (rock, paper, or scissors) defeats one of the other shapes.

**Relation to Finance**: Demonstrates the unpredictability and randomness in some financial decisions, where no single strategy guarantees success.

### Pirate Game

A game where pirates divide gold coins, but they have a strict order of seniority and can vote others off the ship.

**Relation to Finance**: Reflects hierarchical decision-making in financial institutions and the balance between greed and survival.

### Dictator Game

One player (the “dictator”) determines how to split a sum between themselves and another player.

**Relation to Finance**: Examines altruism and fairness in financial decisions, especially in charitable giving or corporate philanthropy.

### Public Goods Game

Players contribute to a public good, which benefits all, but there’s a temptation to free ride on others’ contributions.

**Relation to Finance**: Represents challenges in public finance and the provision of public goods, highlighting the tension between individual (taxpayer) and collective benefits.

### Blotto Game

Players allocate resources across several battlefronts, aiming to win on the majority of fronts.

**Relation to Finance**: Analogous to portfolio diversification, where investors allocate assets across multiple investments to optimize returns.

### War of Attrition

Players decide how long to persist in a costly competition, where the last to quit wins a prize but at a high cost.

**Relation to Finance**: Reflects competitive scenarios in finance where persistence can lead to rewards but at increasing costs, like bidding wars. Or, in the context of geopolitics, the British Empire won two World Wars but at a very high cost, which cost it its status as the world’s top power and the reserve status of the British pound.

### El Farol Bar Problem

Individuals decide whether to go to a bar based on their expectation of how crowded it might be.

**Relation to Finance**: Demonstrates the challenges of predicting market behavior based on collective actions, like anticipating market bubbles or perceiving something as overvalued. If a stock is very popular, does that actually make it less attractive?

### Fair Division

A problem of dividing a resource fairly among several players.

**Relation to Finance**: Relevant in scenarios like inheritance division, joint venture agreements, or any financial scenario requiring equitable distribution.

### Fair Cake-cutting

A specific fair division problem where a cake must be divided among players such that each believes they received a fair share.

**Relation to Finance**: Represents the challenge of subjective valuation in financial assets and what a fair price might be.

### Cournot Game

An economic model where firms compete on the quantity of output produced, affecting the market price.

**Relation to Finance**: Demonstrates oligopolistic competition in financial markets, where major players’ actions influence market prices.

### Deadlock

A situation where two or more players cannot proceed or make decisions, leading to a standstill.

**Relation to Finance**: Reflects situations like stalled financial negotiations or gridlocks in decision-making bodies.

### Diner’s Dilemma

Each diner chooses a costly dish (high benefit) or a cheap dish (low benefit), but all pay the average price. The dilemma is whether to “treat” oneself or save money.

**Relation to Finance**: Highlights the challenges of collective financial decisions, where individual choices impact the group.

### Guess 2/3 of the Average

Players choose a number between 0 and 100, aiming to guess 2/3 of the average of all numbers chosen.

**Relation to Finance**: Demonstrates the challenge of anticipating market behavior based on collective beliefs and actions. Shows the need to think in a multi-order way because the Nash equilibrium of this game is to guess 0 (see FAQ section).

### Kuhn Poker

A simplified poker game designed to capture essential aspects of the real game without complex calculations.

**Relation to Finance**: Represents the simplification of complex financial scenarios to understand fundamental dynamics, like simplified market models.

### Nash Bargaining Game

A two-player game where players negotiate over a divisible good, aiming to maximize their utility.

**Relation to Finance**: Reflects negotiation scenarios in finance, emphasizing mutual benefit and strategic compromise.

### Induction Puzzles

Puzzles that involve players making decisions based on reasoning about other players’ reasoning.

**Relation to Finance**: Demonstrates the complexity of financial decisions where anticipating others’ strategies is important. Who are the other participants in a market, how big are they, and what are they motivated to do?

### Trust Game

One player gives a sum to another, which is multiplied, and the second player decides how much to return.

**Relation to Finance**: Highlights the importance of trust in financial transactions and the potential for mutual benefit through collaboration.

### Princess and Monster Game

A game where a princess hides in a grid, and a monster seeks her. The monster wins if he finds the princess.

**Relation to Finance**: Represents the challenge of finding hidden opportunities or risks in financial markets.

### Rendezvous Problem

Two players try to meet in a location without prior communication.

**Relation to Finance**: Reflects the challenge of synchronizing financial strategies or actions without explicit coordination.

Each of these games offers insights into decision-making, strategy, and human behavior, all of which are crucial in the realms of finance, trading, and investing.

## Theorems in Game Theory & Potential Applications to Finance and Trading

Here are the definitions of the game theory theorems and their applications to finance, trading, or investing:

### Arrow’s Impossibility Theorem

This theorem states that no rank-order electoral system can be designed that always results in a stable and consistent outcome when three or more alternative options are available.

**Application to Finance**: In financial markets, it suggests that no single mechanism can aggregate diverse investor preferences into a consistent market choice, highlighting the challenges of consensus in diverse markets.

### Aumann’s Agreement Theorem

This theorem posits that two players with the same prior probability and common knowledge of each other’s estimates cannot “agree to disagree” about the expected value of a variable (i.e., they should match).

**Application to Finance**: In trading, if two traders have access to the same information and understand each other’s reasoning, they should, in theory, come to the same conclusions about the value of an asset.

### Folk Theorem

In repeated games, the Folk theorem states that any outcome can be sustained as a Nash equilibrium if players are patient enough.

**Application to Finance**: This suggests that in long-term financial interactions, a wide range of outcomes (like pricing or cooperation levels) can be stable, given the right conditions.

### Minimax Theorem

This theorem asserts that in zero-sum games, there exists a strategy for each player that ensures they receive at least their minimax value, regardless of the opponent’s strategy.

**Application to Finance**: In competitive trading scenarios, traders can adopt strategies that minimize their maximum potential loss, ensuring a certain baseline performance.

### Nash’s Theorem

Nash’s theorem guarantees the existence of a Nash equilibrium in finite games.

**Application to Finance**: This suggests that in any financial market scenario with a finite set of strategies, there exists a state where no participant can benefit by changing their strategy unilaterally.

### Negamax Theorem

In zero-sum, two-player games, the value of a position to a player is the negation of the value to the opponent.

**Application to Finance**: This can be applied in hedging strategies where one party’s gain in a financial instrument is offset by another’s loss.

### Purification Theorem

This theorem states that mixed-strategy Nash equilibria in finite games can be approximated by pure-strategy equilibria in perturbed games.

**Application to Finance**: Suggests that complex mixed investment strategies can be approximated by simpler, deterministic strategies under certain conditions. (For example, tactical trading versus simply holding a static asset allocation with periodic rebalancing.)

### Revelation Principle

In mechanism design, the revelation principle asserts that if there’s a mechanism that leads to a certain desirable outcome, there’s a direct mechanism where players truthfully report their private information, leading to the same outcome.

**Application to Finance**: In financial markets, this principle can guide the design of mechanisms (like auctions) to ensure participants reveal their true valuations or beliefs.

### Sprague-Grundy Theorem

In combinatorial game theory, this theorem states that every impartial game is equivalent to a unique nimber, which determines the game’s outcome.

**Application to Finance**: While not directly applicable to most financial scenarios, the theorem’s emphasis on strategy and outcomes can be likened to decision-making processes in trading/investing.

### Zermelo’s Theorem

In finite two-person games of perfect information (like chess), either one of the players has a winning strategy, or both have strategies that lead to a draw.

**Application to Finance**: This theorem underscores the importance of information in financial decision-making, suggesting that with perfect information, market outcomes can be predicted.

## Miscellaneous Concepts in Game Theory & Applications to Finance

### All-pay Auction

An auction in which every bidder must pay regardless of whether they win the item.

**Application to Finance**: This can be likened to competitive markets where firms invest heavily in advertising or R&D, but only one gains the dominant market share. All firms incur costs, but only one reaps the major benefits.

### Alpha-beta Pruning

An algorithm used in tree-searching algorithms to eliminate branches in a decision tree that don’t need to be searched because there already exists a better move available.

**Application to Finance**: Used in algorithmic trading to optimize decision-making processes by ignoring certain trading strategies or assets that are deemed suboptimal.

### Bertrand Paradox

A situation in game theory where two players reach a Nash equilibrium where both charge a price equal to marginal cost, leading to zero economic profit.

**Application to Finance**: Highlights potential pitfalls in competitive pricing in markets, where aggressive price competition can erode profits.

### Bounded Rationality

The idea that in decision-making, the rationality of individuals is limited by the information they have, cognitive limitations, and the finite time they have to make a decision.

**Application to Finance**: Investors and traders can’t process all available information and thus often make decisions based on heuristics, leading to market anomalies.

### Combinatorial Game Theory

A branch of game theory that deals with two-player games with no randomness or hidden information.

**Application to Finance**: Used in analyzing financial strategies in deterministic scenarios, perhaps in structured financial products.

### Confrontation Analysis

A theory and method for analyzing confrontations in terms of dilemmas that parties face in deciding how to act.

**Application to Finance**: Useful in negotiations, such as M&A discussions, where parties weigh the benefits of cooperation versus confrontation.

### Coopetition

A blend of cooperation and competition, where players or firms engage in both competitive and cooperative interactions.

**Application to Finance**: Seen in financial markets where firms might compete in one area but collaborate in another, like joint ventures while competing in other markets.

### Evolutionary Game Theory

A branch of game theory that studies the evolution of strategies in populations over time.

**Application to Finance**: Used to model the evolution of market strategies, especially in algorithmic trading, where strategies evolve based on their success.

### Game Mechanics

The constructs of rules and feedback loops intended to produce enjoyable gameplay.

**Application to Finance**: Used in gamification of financial apps or platforms to engage users and encourage certain financial behaviors.

### No-win Situation

A situation where no player can benefit from any strategy over the others.

**Application to Finance**: Reflects scenarios like market crashes where, due to external factors, almost all long-only investors (e.g., mutual funds, pension funds) face losses and do poorly in absolute terms even when their goal is to outperform a benchmark (even if the benchmark performs even worse).

### Topological Game

A type of game played on topological spaces using the open sets of a topology.

Topological games are complex and hard to describe briefly, so we cover it more in the FAQ section.

**Application to Finance**: While not directly applicable to most financial scenarios, the abstract strategies can be likened to complex financial structures or derivatives.

### Tragedy of the Commons

A situation where individual users, who have open access to a resource, act in their own interest and deplete the resource, harming their own long-term interests.

**Application to Finance**: Seen in scenarios like over-leveraging in financial markets, where individual actions can lead to systemic risks and market crashes. Can also be applied to things like the environment, which also have an impact on markets.

### Tyranny of Small Decisions

A situation where a series of small, individually rational decisions can lead to an outcome that is not optimal or desired for the group.

**Application to Finance**: Reflects scenarios like incremental investment decisions that, over time, can lead to suboptimal portfolio allocations.

## FAQs – Game Theory in Finance

### How is game theory used in finance, trading, and investing?

Game theory in finance, trading, and investing analyzes strategic interactions among market participants.

It helps predict outcomes in competitive scenarios, such as bidding wars, pricing strategies, and merger negotiations.

Traders use it to anticipate other traders’ actions, investors apply it to gauge market reactions to news, and financial institutions employ it to optimize decision-making in regulatory environments.

By understanding the potential moves of others, individuals and institutions can make more informed and strategic financial decisions.

### What is Expected Utility vs. Strategic Dominance?

Expected Utility and Strategic Dominance are two fundamental concepts in game theory.

Here’s a breakdown of each, followed by a comparison:

#### Expected Utility

**Definition**: Expected utility is the sum of the utilities associated with all possible outcomes, weighted by the probability of each outcome. It provides a way for decision-makers to take into account both their payoffs and the likelihood of those payoffs occurring.**Usage**: It is especially useful in situations of uncertainty. Instead of just considering the possible payoffs of a decision, a player also factors in how likely each of those payoffs is.**Basis**: The expected utility theory posits that rational agents should choose the action that maximizes their expected utility. This is foundational in economics and decision theory.

#### Strategic Dominance

**Definition**: In game theory, a strategy is said to dominate another strategy if it always results in a better payoff, no matter what the other players do.**Strict Dominance**: Strategy A strictly dominates Strategy B if A always gives a higher payoff than B, regardless of what other players choose.**Weak Dominance**: Strategy A weakly dominates Strategy B if A always gives at least as high a payoff as B and there’s at least one scenario where A does better.

**Usage**: Dominant strategies are important in simplifying games. If a strategy is dominated (either strictly or weakly), it can often be disregarded in analysis because rational players wouldn’t choose it.**Basis**: Games can sometimes be solved (i.e., equilibrium strategies can be found) by iteratively eliminating dominated strategies.

#### Comparison

**Nature of Concept**:- Expected Utility deals with uncertainty and how players make decisions when the outcomes have probabilistic elements.
- Strategic Dominance deals with comparing strategies based on their payoffs, given the strategies of other players.

**Application**:- Expected Utility is often used in decision-making problems where outcomes are uncertain. For instance, in financial investment decisions, an investor might want to choose the investment that maximizes their expected utility (expected value).
- Strategic Dominance is used in analyzing strategic games to simplify them and identify optimal strategies. For example, in the classic Prisoner’s Dilemma game, the strategy to “betray” strictly dominates the strategy to “stay silent.”

**Rationality**:- Both concepts are underpinned by the idea of rationality. Rational players will choose strategies that maximize their expected utility and will not play strictly dominated strategies.

While both Expected Utility and Strategic Dominance are central concepts in game theory and decision-making, they operate in different realms.

Expected Utility provides a framework for decision-making under uncertainty, while Strategic Dominance offers a method to compare and select strategies in strategic interactions.

### What is the Nash equilibrium of “Guess 2/3 of the Average”?

In the game Guess 2/3 of the Average, the Nash equilibrium is for all players to choose 0.

If all players are rational and are trying to guess 2/3 of the average of all numbers chosen, they will reason as follows:

- If everyone chooses the maximum number, 100, then 2/3 of the average would be 66.67.
- But if everyone thinks this way, then everyone would choose 66.67, and 2/3 of that average would be 44.44.
- Following this logic, the numbers will keep decreasing as players iteratively eliminate higher numbers as possibilities.
- Eventually, the reasoning leads to everyone choosing 0, because if players believe that others will choose 0, then 2/3 of 0 is still 0.

Thus, the Nash equilibrium is for all players to choose 0.

### What are topological games?

A topological game is a type of mathematical game played on a topological space, typically involving two players who take turns choosing points or sets within that space.

The outcome of the game (i.e., which player wins) is determined by the resulting configuration of chosen points or sets, in relation to the topological properties of the space.

One of the most famous topological games is the Banach-Mazur game, where players alternately choose nested closed subsets of a topological space, and the winner is determined based on the intersection of these subsets.

Topological games are used in set-theoretic topology to study properties of topological spaces and to derive results in descriptive set theory.

### What are topological games in simple terms?

In simple terms, topological games are like board games where the “board” is a space with certain properties (like a stretchy surface or a maze), and players take turns making moves on this space.

The rules and outcomes of the game are determined by the shape and structure of the space.

Just as in chess or tic-tac-toe, the goal is to make moves that lead to a winning position, but in topological games, the focus is on the space’s properties and how they affect the game.

### What’s an example of a topological game?

Imagine a game where two players, A and B, take turns drawing dots on a circle.

The rule is simple: You can’t draw a dot too close to another dot (say, closer than an inch).

Player A starts by drawing a dot anywhere on the circle.

Player B then draws another dot, making sure it’s not too close to the first dot. They keep taking turns.

The game ends when a player cannot draw a dot without breaking the “too close” rule. The player who can’t make a move loses.

This is a basic topological game. The circle is our “space,” and the game’s strategy involves understanding this space’s properties (the circle) and predicting where your opponent might draw next.

### How can traders and investors learn from topological games?

Traders and investors can learn from topological games in the following ways:

#### Strategic Thinking

Just as topological games require players to think several moves ahead and anticipate their opponent’s moves, traders and investors need to anticipate market movements and the actions of other market participants.

#### Understanding Complex Systems

Financial markets can be thought of as complex topological spaces where different elements (stocks, commodities, currencies) interact in intricate ways.

Studying topological games can provide insights into navigating such complex systems.

#### Adaptive Strategies

In topological games, a player’s strategy might need to change based on the opponent’s moves.

Similarly, traders and investors must adapt their strategies based on market conditions and external factors.

#### Recognizing Patterns

Topological games often involve recognizing patterns and structures within the space.

In trading, recognizing patterns in price movements, trading volumes, or other indicators can be crucial for success.

#### Risk Management

Just as players in a topological game must consider the risks of each move (e.g., leaving too much space for the opponent), traders and investors must weigh the potential risks and rewards of each investment or trade.

#### Understanding Boundaries and Limits

Topological games often involve boundaries or limits within the space.

Similarly, markets have boundaries (e.g., support and resistance levels in technical analysis) that traders and investors need to consider.

An example might be how the EUR/USD tends to consolidate around 1.00 whenever it reaches that level due to a build-up of orders there.

#### Iterative Decision Making

Topological games often involve iterative decision-making, where each decision impacts future choices.

Similarly, investment decisions made today can impact future opportunities and constraints.

While the direct application of topological games to trading might not be evident, the abstract thinking, pattern recognition, and strategic decision-making skills developed from studying such games can be beneficial for traders and investors.

### How do markets resemble topological spaces?

Markets and topological spaces share similarities in terms of structure, behavior, and interactions.

Here’s how markets can resemble topological spaces:

#### Continuity and Discontinuity

Just as topological spaces can have continuous and discontinuous regions, markets can experience smooth trends or sudden jumps (gaps) in prices.

#### Connectedness

In topology, a space is “connected” if it’s in one piece without gaps.

Similarly, markets can be “connected” when there’s a flow of information, and assets are correlated.

When a significant event affects one asset, it might influence others, showing how they’re interconnected.

#### Boundaries

Just as topological spaces have boundaries, markets have psychological and technical boundaries, like support and resistance levels in asset prices.

#### Compactness

In topology, a space is compact if it’s “bounded” and “closed.”

In markets, certain sectors or asset classes might have a limited range of values they typically trade within, representing a kind of compactness.

#### Local vs. Global Behavior

Topological spaces can have local properties (properties that hold in small regions) and global properties (those that hold across the entire space).

Similarly, markets can have local behaviors (like a stock reacting to company-specific news) and global behaviors (like the entire stock market reacting to macroeconomic news).

#### Complex Interactions

Just as points in a topological space can have intricate relationships and dependencies, assets in a market can interact in complex ways, influenced by a web of economic, political, and social factors.

#### Transformation and Deformation

Topological spaces can undergo transformations that don’t break their inherent structure.

Similarly, markets can adapt and transform in response to external events while maintaining their inherent characteristics.

#### Open and Closed Sets

In topology, spaces are often described in terms of open and closed sets.

In markets, certain periods (like earnings seasons or central bank meetings) can be thought of as “open sets” where new information flows in, while “closed sets” might represent periods of little to no significant news.