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# Game theory uses mathematics to model human decisionmaking in competitive situations. It is ideally suited for analyzing military situations

Game theory is a theoretical framework for conceiving social situations among competing players. According to game theory, the actions and choices of all the participants affect the outcome of each. In some respects, game theory is the science of strategy, or at least the optimal decision-making of independent and competing actors in a strategic setting. The focus of game theory is the game, which serves as a model of an interactive situation among rational players. The key to game theory is that one player’s payoff is contingent on the strategy implemented by the other player. The game identifies the players’ identities, preferences, and available strategies and how these strategies affect the outcome. Depending on the model, various other requirements or assumptions may be necessary.

Some of the few terms commonly used in the study of game theory are: Game is Any set of circumstances that has a result dependent on the actions of two or more decision-makers (players); Players: A strategic decision-maker within the context of the game Strategy: A complete plan of action a player will take given the set of circumstances that might arise within the game; Payoff: The payout a player receives from arriving at a particular outcome (The payout can be in any quantifiable form, from dollars to utility.) ;Information set: The information available at a given point in the game (The term information set is most usually applied when the game has a sequential component.); Equilibrium: The point in a game where both players have made their decisions and an outcome is reached. Nash Equilibrium is an outcome reached that, once achieved, means no player can increase payoff by changing decisions unilaterally. It can also be thought of as “no regrets,” in the sense that once a decision is made, the player will have no regrets concerning decisions considering the consequences.

Although there are many types (e.g., symmetric/asymmetric, simultaneous/sequential, et al.) of game theories, cooperative and non-cooperative game theories are the most common. Cooperative game theory deals with how coalitions, or cooperative groups, interact when only the payoffs are known. It is a game between coalitions of players rather than between individuals, and it questions how groups form and how they allocate the payoff among players. Non-cooperative game theory deals with how rational economic agents deal with each other to achieve their own goals. The most common non-cooperative game is the strategic game, in which only the available strategies and the outcomes that result from a combination of choices are listed. A simplistic example of a real-world non-cooperative game is Rock-Paper-Scissors.

The biggest issue with game theory is that, like most other economic models, it relies on the assumption that people are rational actors that are self-interested and utility-maximizing. Of course, we are social beings who do cooperate and do care about the welfare of others, often at our own expense. Game theory cannot account for the fact that in some situations we may fall into a Nash equilibrium, and other times not, depending on the social context and who the players are.

Game theory has a wide range of applications, including psychology, evolutionary biology, war, politics, economics, and business. Despite its many advances, game theory is still a young and developing science.

### Applied Game Theory to Improve Strategic and Tactical Military Decisions

In 1950, Haywood proposed the use of game theory for military decision making while at the Air War College. This work culminated in an article, ‘Military Decisions and Game Theory”. In 2003, LTC Cantwell proposed a methodology using zero-sum games to improve military decision making in choosing courses of action. He proposed using ordinal values to fill in the zero-sum payoff matrix for the USA and the opponent and then solve the game. Cantwell  showed and presented a ten step by step procedure to assist analysts in comparing courses of action for military decisions.

He illustrated his method using the Battle at Tannenberg between Russia and Germany in 1914 as his example.

Cantwell’s ten step procedure  was presented as follows:
Step 1: Select the best-case friendly course of action for the friendly forces that achieves a decisive victory.
Step 2: Rank order all the friendly courses of action from best effects possible to worse effects possible.
Step 3: Rank order the enemy courses of action from best to worst in each row for the friendly player.
Step 4: Determine if the effect of the enemy courses of action result in a potential loss, tie or win for the friendly player in every combination across each row.
Step 5: Place the product of the number of rows multiplier by the number pf column in the box representing the best case scenario for each player.
Step 6-9: Rank order all combination for wins, tie, and loses escending down from the value of Step 5 to 1.
Step 10: Put the matrix into a conventional format as a payoff matrix for the friendly player.

### Using Game Theory to Analyze Operations Against Time-Critical Targets

When planning operations against time-critical targets (TCTs), military commanders typically think about how much capability they need to kill enemy forces. However, they also consider how their strategies will affect the enemy’s behavior. TCT operations include suppression of enemy air defenses (SEAD), interdiction of moving forces, and attacks against theater ballistic missiles (TBMs). Convincing an enemy not to fire surface-to-air missiles (SAMs), not to move his forces, or not to launch TBMs is often a satisfactory short-term alternative to physically destroying enemy systems.

A study by RAND Project AIR FORCE (PAF) shows how military planners can use game theory to understand the effects of U.S. strategy and capabilities on the enemy in TCT operations. Game theory uses mathematics to model human decisionmaking in competitive situations. It is ideally suited for analyzing military situations because it depicts the realistic situation in which both sides are free to choose their best “moves” and to adjust their strategy over time. Military planners can apply these principles to TCT operations through game theoretic analysis. The method consists of the following steps:

• Determine the tactical options available to each side. For example, in a simple SEAD operation, the attacker can choose to fly a strike aircraft or a SEAD aircraft. The defender may choose to activate SAMs or to leave them inactive.
• Assign a numerical value to each possible outcome. Analysts represent commanders in the field by judging the potential gain or loss of an exchange. These numbers reflect real-world measurements such as the strength of a weapon system and the probability of hitting a target.
• Calculate all possible strategies and their outcomes. Intelligent opponents vary their tactics in order to appear unpredictable to the enemy. Thus, a combatant’s overall strategy is determined by how often he chooses one tactical option over another. Analysts calculate all possible strategies and the net gain or loss to each side.
• Find each side’s optimum strategy. Experience teaches that as opponents in a game adjust to each other’s actions, each player will eventually settle on an optimum strategy. In military terms, the optimum strategy is not necessarily the most desirable outcome (i.e., winning the exchange), but the best that one can do against an opponent of given strength.
• Determine the expected result of the game. Having found each side’s optimum strategy, analysts check to see whether the outcome of the encounter favors the attacker or the defender.

Game theoretic analysis enables analysts to see how an intelligent opponent is likely to behave in a given situation and which side is likely to win. If both sides correctly ascertain the situation, then the losing side may decide not to participate. For example, in a simple SEAD encounter, the defender might decide that preserving his SAMs is more important than attempting to shoot down strike aircraft if his chances of inflicting heavy losses on the attacker are small. Insights such as these help military planners to understand how much capability they would need to achieve the best outcome for their side.

### Game-theory research better allocates military resources, fight cancer

U.S. Army game-theory research using artificial intelligence may help treat cancer and other diseases, improve cybersecurity, deploy Soldiers and assets more efficiently and even win a poker game. New research, published in Science, and conducted by scientists at Carnegie Mellon University, developed an artificial intelligence program called Pluribus that defeated leading professionals in six-player no-limit Texas hold’em poker.The Army and National Science Foundation funded the mathematics modeling portion of the research, while funding from Facebook was specific to the poker.

“It’s all about strategy,” said Dr. Purush Iyer, division chief, network sciences at the Army Research Office, an element of the U.S. Army Combat Capabilities Development Command’s Army Research Laboratory. “A limiting factor in game theory has always been scalability (i.e., ability to deal with exponentially increasing state space). Poker is an accessible example to show how these mathematical models can be used to devise strategies for situations where a person doesn’t have complete information — they don’t know what the adversaries will do, and what their capabilities are.” This research is extremely relevant to many real-world and military challenges that involve multiple parties such as cybersecurity and defense posturing, he said.Poker has been an AI challenge because it is an incomplete information game, where players cannot be certain which cards are in play and opponents can, and will, bluff, much like military strategy.

“Thus far, superhuman AI milestones in strategic reasoning have been limited to two-party competition,” said Dr. Tuomas Sandholm, Angel Jordan Professor of Computer Science, who developed Pluribus with Noam Brown, who is finishing his doctorate in Carnegie Mellon’s Computer Science Department as a research scientist at Facebook AI. “The ability to beat five other players in such a complicated game opens up new opportunities to use AI to solve a wide variety of real-world problems.” “Playing a six-player game rather than head-to-head requires fundamental changes in how the AI develops its playing strategy,” said Brown, who joined Facebook AI last year.

Pluribus dispenses with theoretical guarantees of success and nevertheless develops strategies that enable it to consistently outplay opponents. Pluribus first computes a blueprint strategy by playing six copies of itself, which is sufficient for the first round of betting. From that point on, Pluribus does a more detailed search of possible moves in a finer-grained abstraction of game. It looks ahead several moves as it does so, but not requiring looking ahead all the way to the end of the game, which would be computationally prohibitive. Limited-lookahead search is a standard approach in perfect-information games, but is extremely challenging in imperfect-information games. A new limited-lookahead search algorithm is the main breakthrough that enabled Pluribus to achieve superhuman multi-player poker.

The software also seeks to be unpredictable. For instance, betting would make sense if the AI held the best possible hand, but if the AI bets only when it has the best hand, opponents will quickly catch on. So Pluribus calculates how it would act with every possible hand it could hold and then computes a strategy that is balanced across all of those possibilities. With Army funding, Sandholm and some of his other students are developing related techniques for bio-steering, where the researchers are computing optimal treatment plans that steer a patient’s immune system to better fight cancers, autoimmune diseases, infections, etc.

Previous Army-funded game theory research is now being used by the Transportation Security Administration, the U.S. Coast Guard and the Los Angeles Metro Rail to schedule resources in a manner that decreases cost for the those organizations ensuring safety while increasing the costs for an adversary, thus reducing the chances for attacks. Furthermore, Army-funded foundational research in algorithmic game theory has been used in civil society to reduce poaching of elephants in Queen Elizabeth Forest, Uganda, and tigers in Southeast Asia, as well as in addressing homelessness and implementing HIV-prevention campaigns in Los Angeles.

“The research work of Dr. Sandholm and others will be used in a variety of ways in the not-too-distant future to address societal problems in a cost-effective manner,” Iyer said. “Dr. Sandholm’s work is an exciting advance in game-theory; the applications are enormous.” The CCDC Army Research Laboratory (ARL) is an element of the U.S. Army Combat Capabilities Development Command. As the Army’s corporate research laboratory, ARL discovers, innovates and transitions science and technology to ensure dominant strategic land power. Through collaboration across the command’s core technical competencies, CCDC leads in the discovery, development and delivery of the technology-based capabilities required to make Soldiers more lethal to win our Nation’s wars and come home safely. CCDC is a major subordinate command of the U.S. Army Futures Command.

### The project  “Multi-Scale Network Games of Collusion and Competition.”

Shedding light on how officials at different levels of government can work together to maximize COVID-safe behavior is a new goal of a multiscale game theory project funded with \$6.5 million from the Department of Defense. Game theory and COVID-19: Major defense project pivots to explore how to coordinate safe behavior. Mingyan Liu, leader of the project and the Peter and Evelyn Fuss Chair of Electrical and Computer Engineering, presented her team’s work at a recent “Call to Arms” virtual conference, held by the National Science Foundation’s Networking Technology Systems group.

When human behavior is competitive, we don’t use resources in the way that is most efficient for the community—as seen in behaviors like mask, sanitizer and toilet paper hoarding. But most of our decisions about how to behave aren’t entirely individualistic. We make them as part of a community. We are swayed both by leadership—and the incentives and disincentives that they can offer—as well as altruism. Most of the literature in game theory examines individual behavior, but Liu and her colleagues are exploring what happens when decisions are made at multiple scales. This is particularly relevant during the current COVID-19 pandemic, when decisions are made by individuals, local governments, state governments and nations. Do we act for the common good, or do we do what we perceive as serving our individual interests?

“An example, in the present context, is someone who ignores shelter-at-home orders and essentially benefits from other people’s decision to comply with the order. This is what’s called freeriding,” Liu said. Freeriders are protected from the virus through the decrease in transmission brought about by everyone who stays at home. This principle also applies to local and state officials making their own calculations about mitigating the risk of COVID-19. One state with a high COVID-19 case load may decide to shut down, while a neighboring state with a smaller case load may have the advantage of staying open, again benefiting from the free rider effect.“The global pandemic is the most salient threat we face at the moment, ” said Purush Iyer, program manager at the Army Research Office, an element of the U.S. Army Combat Capabilities Development Command’s Army Research Laboratory.

“While the U.S. Army’s interest in network games includes understanding the impact of the adversarial groups in a host population, electronic warfare, and distributed weapon systems, we fully support exploring the impact of measures to control the spread of disease.” To begin with, the team is exploring how to model compliance or lack of compliance regarding COVID-19 orders and recommendations in their game-theory framework. The protective behaviors include not going out, wearing a mask when going out, and handwashing and sanitizer use when returning from being out.

The factors that may influence compliance often include the prevalence of COVID-19 in the local community, a person’s vulnerability or proximity to vulnerable individuals, and general awareness. But they may also be affected by the timing of the order and even the words and phrases chosen to give the justification and restrictions. This aspect of the analysis will allow the team to then investigate communitywide behavior as a result of high-level policies. Liu plans to connect behaviors identified from such data with COVID-19 case data to discover which restrictions and recommendations are most effective. “We’re also interested in understanding what additional mechanisms or policies could be introduced to make the overall system more efficient—for instance, enabling more collaboration among communities rather than competition,” Liu said.

She cited the way that states are currently fighting one another for federal supply of medical equipment even as some come together on a plan to begin reopening the economy. For now, Liu’s team is best equipped to model strategic decisions associated with social distancing at the individual and community levels, but they have plans to include economic concerns as well. The state that is able to remain open because its neighbors are closed is a free rider in the sense of limiting virus spread, but it may also play an important economic role in manufacturing and distribution, helping to head off shortages.