Summary of Findings (White Team): Game Theory ( 3.5 out of 5 Stars)

Note: This post represents the synthesis of the thoughts, procedures and experiences of others as represented in the 8 articles read in advance (see previous posts) and the discussion among the students and instructor during the Advanced Analytic Techniques class at Mercyhurst University in March 2013 regarding Game Theory specifically. This technique was evaluated based on its overall validity, simplicity, flexibility and its ability to effectively use unstructured data.


Description:
Mesquita (2011) defines game theory as a "a body of reasoning, grounded in mathematics but readily understood intuitively as a reflection of how people may behave, particularly in situations that involve high stakes for them. It is part of a family of theories that assume people are rational, meaning that they do what they believe (perhaps mistakenly) is in their best interest." Game theory is a methodology that allows a player to make a decision based on predictions of what the other player will do and weigh their potential options in relation to the options of the other players. Each payoff will be different depending on the combination of strategies.


Strengths:
1. Looks at multiple scenarios between the actions of two parties
2. Applicable to many fields
3. Denial and deception actions by the adversary can be taken into account within game theory models
4. Not entirely accurate or helpful if only performed once
5. Is effective in emphasizing the importance of critical thinking in making a decision and understanding the potential consequences of particular decisions


Weaknesses:
1. Have to be able to play a role
2. Level of accuracy can be low
3. Difficult to simulate unless it is a real-life situation
4. Assumes that the players are rational actors
5. Some scenarios need to be played multiple times to witness a discernible pattern of behavior between the parties.
6. Rational behavior within the game model is different than rational behavior with human interactions. Human interactions should be taken into account with the game model.
7. An iterated game will change the result of the outcome


Step by Step Action:
1. Construct a matrix
2. Determine possible decisions the players are likely to make
3. Determine how the players maximize the benefits based on those decisions
4. Conduct the exercise to see how the possible outcomes play out
5. Repeat exercise to get better results


Exercise:
Begin with 20 gold coins considered your “loot”. Each person playing the game is assigned a number. The person with the highest number holds the most power. The person holding the most power begins the game by making a proposal as to how to assign the coins, keeping in mind the person with the most power wants maximize the amount of coins they have while appeasing a majority of the players. A vote is taken, a majority is necessary for the person with the highest number to keep power. If a majority is not achieved power moves down to the next person and the process starts again.

Some interesting dynamics resulted when we calculated the exercise as a class. Rationality for the first round demonstrated that the rationality to distribute the coins evenly among the participants was not the chosen course of action by the first offering which was voided. Playing the game multiple times demonstrated the need for the first two individuals to give offerings that were fair to everyone, and most significantly allow the players with the most power to gain coins ( positive sum games) instead of zero sum games in which the other players benefitted from the higher ranked individuals incorrect rationality.


Source:

Mesquita, B (2011) Applications of Game Theory in Support of Intelligence Analysis. Intelligence Analysis: Behavioral and Social Scientific Foundations, 57-82. Retrieved from http://www.nap.edu/openbook.php?record_id=13062&page=57

Summary of Findings (Green Team): Game Theory (2.875 out of 5 Stars)

Game Theory

Green Team

Rating (2.875 out of 5 Stars)

Note: This post represents the synthesis of the thoughts, procedures and experiences of others as represented in the 8 articles read in advance (see previous posts) and the discussion among the students and instructor during the Advanced Analytic Techniques class at Mercyhurst University in April 2013 regarding Game Theory specifically. This technique was evaluated based on its overall validity, simplicity, flexibility and its ability to effectively use unstructured data.

Description:

Bruce Bueno de Mesquita defines game theory as “ a body of reasoning, grounded in mathematics but readily understood intuitively as a reflection of how people may behave, particularly in situations that involve high stakes for them. It is part of a family of theories that assume people are rational, meaning that they do what they believe (perhaps mistakenly) is in their best interest.” The method attempts to identify potential actions and reactions from players using a variety of factors, the most important of which is the belief that players will act in their best interest. Additionally, this theory maintains the understanding that individuals will make rational decisions and that rationality will not necessarily end with the most optimal solution for both (or more) parties.

Strengths:

• Can use and evaluate a range of unstructured data
• Can be used in many fields including economics, international relations, and intelligence
• Some models of game theory are simple, though models increase in sophistication and difficulty
  
  

Weaknesses:

• Relies heavily on individuals acting in their best interests - does not account for altruistic behavior
• Single play games are not very useful and in reality, games/scenarios cannot or will not always be repeated
• Complex applications of game theory require substantial knowledge of math and statistics
• Context heavy, relying on isolated, hypothetical scenarios
• Operates under a number of assumptions, including actors’ motivations and psyches
  
  

How-To:

1. Find a problem that can be modeled through game theory (nothing overly complex with too many factors).  Make sure the simplified essence of the problem is used.  
2. List all actors and options.  A table is the most common representation technique -- since it visually portrays a payout matrix
3. Assign values to each of the potential outcomes.
4. Weigh potential outcomes according to these values.
5. Describe situations in which these outcomes are likely to occur and then choose the most likely, regardless if it is the ‘best’ outcome or not.
   
   

Personal Application of Technique:

The class was tasked with finding the optimal outcome for themselves in a pirate puzzle application of game theory. In this application of the pirate puzzle, five pirates have an understanding of rank that is respected on the ship. The pirates must decide how to divvy up a bounty of twenty gold (chocolate) coins. Players operated under the assumption that they wanted the twenty gold coins.  The highest ranking pirate makes the initial proposal to divide the coins, knowing he must gain majority’s approval in order for the proposal to pass. In our example, the pirate who fails to gain the majority vote is out of the running for the gold coins and the next highest ranking pirate makes his own proposal until one proposal is agreed upon. In the event of a tie, the highest ranking pirate involved in the proposal casts the deciding vote. The game was run twice in order to increase the presentation of different options.

In both instances, the highest ranking pirate in the class failed to gain the majority vote and was eliminated from the running. Instead, in the first instance, the game was played out to the third ranking pirate. The next iteration ended with the second highest ranked pirate, who was able to appeal to the lowest ranked pirate to earn a fifty-percent vote. Despite what game theory suggests, the lowest ranked pirate in our scenario was not satisfied with an offer of a single coin, or a few coins, even knowing she could be left with none if she let game continue. During the second round of the game, the individuals acted more to their interests rather than attempting to appease everyone.

Rating:  2.875 out of 5 stars

For Further Information:

Stewart, I. (1999). Mathematical recreations: A puzzle for pirates. Scientific American, 98-99. Retrieved from http://www.cse.iitb.ac.in/~saifhhasan/files/pirate_puzzle.pdf

Source: Mesquita, B (2011) Applications of Game Theory in Support of Intelligence Analysis. Intelligence Analysis: Behavioral and Social Scientific Foundations, 57-82. Retrieved from http://www.nap.edu/openbook.php?record_id=13062&page=57

Relations between Free Trade and Economic Protection: A Game Theory Analysis

Relations between Free Trade and Economic Protection: A Game Theory Analysis

Summary:

Lin and Lee (2012) state that there is a conflict between mutual policies between countries over discussions that result in a stalemate between free trade and environmental protection issues.  The authors' goal of their research endeavor was to find actions in which both parties can pursue both policies of free trade and environmental protection that do not harm both sides and can result in positive-sum games for the parties involved.  Thus, Lin and Lee (2012) had the purpose of using game theory to: "examine the possibility of including environmental issues into trade negotiations and examining the impasse confronting the coordination between trade and environmental protection (592)."

The results of the study demonstrated that once trade and environmental liberalization were brought in together within negotiations, the international community was more likely to combine both trade and environmental issues into the decision-making process.  Overall, this finding benefited developed nations interests' and forced developing nations to join in the discussions or be left out all together.  Moreover, the developing nations seem to stick to their own agenda and have different priorities as compared to the developed nations in terms of trade and economic polices.  The authors state that the extreme differences between the North and the South countries disturb the balance between trade and environmental policy negotiations (Lin and Lee, 2012).

Through various Prisoner Dilemma's (PD),  Lin and Lee (2012) examined various issues that pertained to trade agreements and environmental protection, which can be witnessed by the graphic above.  Through the utilization of the above PD, the authors came to the following conclusions for developed and developing nations.  Developed nations believe that trade restrictions were the most powerful decision-making option to solve environmental issues against the second party.  Environmental protection is a top priority in terms in conducting international trade.  Lastly, the developed nations were more likely to strive for unification of international environment standards as compared to developing nations (Lin and Lee, 2012).

The PD emphasized that developing nations view economic development and the elimination of poverty as their top challenges.  In addition, developing nations affirm that tariffs known as green barriers which are created for environmental protection, negatively affect the developing nations export economy.  The developing nations can not meet the environmental standards initiated by the developed nations due to limited financial resources.  This aspect is likely to increase production costs in developing nations, but also will reduce the amount of exports.  Lastly, the developing nations see it as unethical that the developed nations ban their products from being exported into their country, but consistently move their environmentally harmful production facilities into the developing countries around the globe (Lin and Lee, 2012).

Critique:

It was interesting how the authors utilized game theory to examine the possibility of including environmental issues in trade negotiations between developed and developing nations.  The use of PD displayed some interesting characteristics of the differences in decision-making and viewpoints as it curtailed to trade and environmental negotiations between the viewpoints of developed and developing nations.  The findings presented in this study would be able to display to decision-makers the differing viewpoints between developed and developing nations and applying this knowledge to better serve the needs of the two parties.  Hence, creating positive sum games.  This knowledge may be able to provide decision-makers the ability to likely increase the effectiveness of negotiations that bring in environmental factors into trade agreement conversations.

However, the study needs to elaborate further what changes to the PD would create the scenario for developing nations to become more willing to participate in trade negotiations that bring in environmental issues.  More in-depth utilization of PD would allow the authors to determine what scenarios made developing nations stick to their own criteria/decisions and how could those scenarios be reversed to allow for mutual cooperation between the parties. Overall, this study brings up intelligence gaps that could be further researched to provide possibly more actionable intelligence for a decision-maker who deals in negotiations of trade and environmental issues between nations of differing economic standing.

Source:

Lin, C.M. & Lee, C.K. (2012). Relations between free trade and economic protection: a game theory analysis. International Journal of Management, 29 (2), 591-605. Retrieved from http://ehis.ebscohost.com/ehost/detail?sid=6e2f16a5-67b2-478f-b717-fad8b32698c0%40sessionmgr10&vid=1&hid=6&bdata=JnNpdGU9ZWhvc3QtbGl2ZQ%3d%3d#db=bsh&AN=76442241. 

Applying Analytical Methods to Study Terrorism

Summary:

In Applying Analytical Methods to Study Terrorism, Sandler and Enders discuss choice-theoretic and game-theoretic methods and the study of trends and forecasting in terrorism. They apply game-theory as a method to the study of terrorism in which "strategic behavior among rational players is assumed that allows a player to anticipate the response of others to its own actions". Operating under the assumption that terrorists function as rational agents, game theory has the potential to allow us to make predictions and decisions in response to their threats and attacks. It also allows for strategic advantage and bargaining behavior for decision-makers, a behavior relevant to terrorist organizations. Additionally, choice-theoretic models allow for a subject to maximize their profit (i.e. greatest reduction in terrorism) and forecast terrorism patterns.

Data on trends of terrorist attacks is provided in context, concluding that as rational actors, terrorists will reduce risk by choosing the mode of attack with least risk: bombings. With additional statistical analyses, studies have shown that a rise in casualties associated with transnational terrorism began with the U.S. embassy takeover in Tehran and the Soviet invasion of Afghanistan in 1979. Trends in the cycles of threats versus incidents with deaths from terrorist attacks can also be evident. "Threats had a primary cycle of just under year, while incidents with deaths had a primary cycle of almost 10 years." Understanding these cycles aids in the decision-making for when to augment defenses. Intervention analysis is another statistical technique used in counterterrorism that measures unintended consequences of policies on terrorist attacks internationally.

The authors state that "game-theory is an appropriate tool for understanding the strategic interactions associated with terrorists and those charged with counterterrorism." They proceed to offer a game-theory matrix involving two countries who both have the following decisions: preempt a terrorist attack, maintain the status quo and do nothing, or defend against a terrorist attack. The matrix combines two 2 x 2 matrices together. This figure is shown below.

According to this matrix, the Country A would gain the most by doing nothing if Country B preempts and therefore allowing Country A to "free-ride" on their decision and in essence gaining from the offensive attack (top, left 2 x 2). This is reliant on the other country actually preempting an attack. However, if the roles of costs and benefits are switched, then each country's dominant strategy is to defend rather than maintain the status quo, but this has the lowest payoff in the game (bottom, right 2 x 2). Thus, this leads to an overly defended but unsafe world.

Critique:

The provided data and statistical analyses on transnational terrorism was helpful in providing a context and additional background information on the decision-making skills of terrorists. This is important when applying the method of game-theory to this context because there first must be an understanding of the subjects in the method. The authors are also effective in explaining the benefit of game-theory's application to counterterrorism and showing how terrorists can exploit asymmetric warfare, evident by the matrix.

A serious limitation in this study, however, which was noted by the authors is that the matrix does not include terrorists as players. Although this issue was highlighted in the article, the authors did not attempt to include terrorists in the game-theory model they presented. This would require a new three-agent game including two countries/ governments and terrorists; it would have been nice to see an attempt at somehow creating such a matrix in the article. Because a country's decision on how to proceed in counterterrorism relies heavily on the behavior of the terrorist organizations, including them as a player in the game would add new beneficial insight into the methodology for this field. Nevertheless, the authors did apply a double matrix in which country A and B had two likely scenarios and weighed the cost/ benefit of the two decisions in relation to each other which was interesting and valuable.

Source:

Sandler, T., & Enders, W. (2007). Applying analytical methods to study terrorism. International Studies Perspectives, 8, 287-302. Retrieved from http://ehis.ebscohost.com/eds/pdfviewer/pdfviewer?sid=312e4b6a-1c25-4322-8422-638d3ebe16fa@sessionmgr10&vid=2&hid=17

Applications of Game Theory in Support of Intelligence Analysis

Bruce Bueno de Mesquita wrote an article in Intelligence Analysis: Behavioral and Social Scientific Foundations which examined how game theory can be applied to intelligence analysis.

Mesquita begins by defining game theory as "a body of reasoning, grounded in mathematics but readily understood intuitively as a reflection of how people may behave, particularly in situations that involve high stakes for them. It is part of a family of theories that assume people are rational, meaning that they do what they believe (perhaps mistakenly) is in their best interest." Mesquita discusses the basics of game theory, such as anticipating how others will act, constraints, considering counter-factual actions, and using cost-benefit analysis to help make decisions. Mesquita argues that game theory helps integrate knowledge based on different theories, such as structural, organizational, behavioral, and psychological.  

In the realm of national security, Mesquita discusses five constraints that effect analysis. These are uncertainty, risks, distribution of costs and benefits, coordination, and patience. According to Mesquita, game theory examines uncertainty in two different ways: random shocks to a situation and not knowing a critical piece of information about a player. Random events can change the expectations of a player, leading to a change in action. Examples can be a key figure dying or an earthquake, both events of which can alter the focus of a decision maker. Not knowing about the player also increases uncertainty, as it is difficult to determine what game players are playing. 

Risks deal with the probability of alternative results which arise from different choices of action. Part of what makes risks so important is how players respond to risk. Some are more prone to taking risks while others are less prone. Mesquita gives an example of risk while discussing the Shah of Iran. Nondemocratic leaders who stay in office past two years see a significant year-to-year decline in the risk of being forcibly removed from office, as long as all else is equal. This is why it was a surprise when the Shah was ousted after over 20 years after his coronation. What was not taken into account was the Shah's terminal illness, which tend to increase the risk of a nondemocratic being deposed.

Distribution of costs and benefits deals with the outcomes of games. This leads to players bluffing in order to gain their desired outcome. There are two types of bluffing: cheap and costly. A cheap bluff is a bluff that costs the player nothing, such as rhetoric or hinted threats in official communiques. A costly bluff, which is more likely to be noticed by the other player or players, could be the mobilization of military forces or missile tests. The more costly the bluff, the more likely the other players will believe it.

Coordination is a problem that arises when players work towards a common goal or resolution of an issue. There are two types of coordination issues: pure coordination problems and more complex coordination problems. A pure coordination problem is deciding which side of the road allied tanks should drive on. A more complex coordination problem is creating incentives to get allies to coordinate in a time of war. En example of this could be promising an ally territory from the enemy if they were to invade at a certain time.

Lastly, patience examines the value a given cost or benefit has today when compared with tomorrow (or longer). The more patient an decision maker is, the closer the future cost or benefit is to the current one. The more impatient the person, the greater the difference in current and future costs and benefits.

Mesquita concludes by discussing some of the limitations of game theory. Game theory makes very heavy assumptions about information and people. For a game to work, it requires that some critical information is common knowledge (Player A knows that Player B has nerve gas, and Player B knows that Player A knows). Same applies with people: Player A assumes that Player B will continue playing the game, or assumes that there will be no more players. Lastly, game theory models can lead to very (sometimes overly) precise outcomes. This can cause issues as it can be difficult to be adaptive with outcomes

Critique

Mesquita's article was very readable, which is an issue that I have come across when attempting to learn more about game theory. In particular, his caveat about rationality in game theory was right on the spot: what may be irrational to some is rational to others. He explained concepts very well and gave great examples of the different types of constraint. That being said, there were two issues that I can see some having with his discussion of game theory and intelligence.

First, at no point did Mesquita use mathematical formulas. While this was not an issue with me, as I would not have understood them anyways, it could be problematic to those that want to know how to do game theory mathematically. The purpose of the article was not to teach the reader how to crate game theory models, but to explain how it can be applied to intelligence. Even so, it could have been useful to those who are interested in creating a game theory model.

Second, while his discussion of game theory was interesting and the examples he used helped explain the concepts, most of it seemed to be more focused on international relations than intelligence. Granted, game theory is heavily used in international relations, so it would make sense for the majority of examples to be international relations-focused. The constraints he discussed definitely have applications in the intelligence field, but this application seemed to be a secondary objective.

Source:  Mesquita, B (2011) Applications of Game Theory in Support of Intelligence Analysis. Intelligence Analysis: Behavioral and Social Scientific Foundations,  57-82. Retrieved from http://www.nap.edu/openbook.php?record_id=13062&page=57



  

Game Theory-Based Identification of Facility Use Restriction for the Movement of Hazardous Materials Under Terrorist Threat

Summary:
Reilly, Nozick, Xu, and Jones (2012) developed a model of interactions among government, terrorists, and carriers of hazardous materials using game theory. Their intention was to understand how governments might prohibit certain travel routes for carriers shipping hazardous materials, how the carrier might decide which routes to take in response to the prohibitions and the threat of terrorism, and how terrorists might target available links and in what frequency. An extension of a two-person, non-zero sum game, Reilly et al. constructed a  non-cooperative, non-zero sum three-person game in which the government is the leader and both the carriers and terrorists are followers.

The idea of the research is that governments will respond to threat levels of terrorist activities by restricting the transportation of hazardous materials that could place the greater population at risk. These restrictions would likely come in the form of prohibited travel routes for carriers to reduce risk. In reaction to these restrictions, carriers must decide which routes to travel while considering travel time and consequence measure, a combination of population exposure and accident probability. Terrorists meanwhile react to government restrictions by choosing targets whose access will not be impeded by such route restrictions. The research operates under the assumption that the terrorists will be equally aware of route restrictions as carriers will be.

A case study applying this to the rail systems used by carriers of hazardous materials found 259 links which could be considered of interest to the government in terms of risk restriction. The research considers the change in expected payoff for terrorists upon restrictions, compared to the change for carriers. In some surprising cases, the expected payoff for terrorists increases with government restrictions, while other times it returns to the same point as no restrictions, though at a substantial expected loss for carriers who cannot transport a percentage of their total carloads due to the restrictions. This shows that despite government's best intentions, route restrictions may further exacerbate threats.

Critique:
The research by Reilly et al. only represents the interactions between the three parties for the movement of hazardous materials by a single carrier. While this creates reasonable rules to predict carrier and terrorist actions with regards to a maximum allowable expected payoff for the terrorist, it significantly limits the scope and utility of the research in the intelligence and policy fields. As the authors note, to improve the research the formulation and solution procedure should be expanded to handle multiple carriers with several origins and destinations. Additionally, the current research lacks depth in that it only considers single attacks by terrorist organizations, rather than coordinated attacks. Given the maximum profit-seeking nature of terrorists, this inclusion would also substantially improve the utility of the research.

The research, or perhaps the limitation of game theory, also fails to include the albeit unlikely scenario that government restrictions on particular routes would cause either carriers to cancel shipments altogether and/or for terrorists to change targets away from hazardous materials if the expected value is not significant enough. In some of the cases presented in the findings, the carriers project substantial loss, though this application of game theory does not account for the potential political backlash carriers may inflict on government, complicating the goals of government. It is narrow-minded to assume that any government will strictly look to minimize security threats without considering the economic and political backlash such closures would have. I do not consider this a failure of this particular research, rather a weakness of game theory itself.

Source: Reilly, A., Nozick, L., Xu, N., and Jones, D. (2012). Game theory-based identification of facility use restriction for the movement of hazardous materials under terrorist threat. Transportation Research Part E, 48(1), 115-131. Retrieved from http://www.sciencedirect.com/science/article/pii/S1366554511000810

Applications of Game Theoretic Reasoning to Basketball Situations

Introduction

Game theory is a specific model of thinking that uses precise logic reasoning to solve problems. It is used to help understand situations in which decision-makers interact. Osborne (2000) provides an excellent introduction to game theory, focused more on the ability to follow logical processes than mathematical reasoning. Game theory involves using models to simplify transform reality into an abstraction which is more easily analyzed or understood. Most models are based on a set of actions available to decision makers, and assume that decision makers are rational actors, meaning that they will always choose the most preferred action. Game theory works best with a finite action space and a specific set of rules for the environment.

Basketball is an excellent example of a zero-sum game, with two actors, or teams, both pursuing the same goal, victory. This is zero-sum because one team’s victory means the other team’s failure. In these game’s the coach’s only objective is to win, but he or she must juggle a number of factors to do so. Certain situations within basketball provide great opportunities for game theoretic analysis. In this paper, two of these will be evaluated. In the first, I look at the classic issue of foul trouble. Conventional wisdom is that players should be immediately benched, but using game theoretic principles, this assumption can be questioned. In the second situation, I modeled an end of game situation where one team trails another by two points with less than 20 seconds left in the game. This time crunch shrinks the options available to either coach, and forces them to choose to shoot or defend, respectively, the two or three point shot. I assigned payoffs and calculated a mixed strategy equilibrium for both the offensive and defensive teams.

Situation 1: Foul Trouble

1: Total threshold fouls and yanks per team 2006-2007 to 2009-2010. 

Teams are consistent regarding benching players in 

foul trouble. Source: (Maymin, Maymin, & Shen, 2012)

Star players often give the team a better chance to win, and their playing time should be maximized. One factor which reduces the playing time is fouls. In the National Basketball Association (NBA), a player can commit at most six personal fouls before he is disqualified from the game. Referees govern fouls according to the rules set down by the NBA. Coaches may sit their star players for an extended period of time if they feel that their number of fouls puts them at risk for disqualification from play. Maymin et al. discuss this problem from a resource allocation standpoint that addresses strategic idling. “The advantage of yanking is that the starter will likely be able to play at the crucial end of the game but the disadvantage is that he may not play as many minutes as he otherwise would. On the other hand, if a starter is kept in the game, he may not play at his full potential, as the opposing team tries to induce him to commit another foul” (Maymin, Maymin, & Shen, 2012). Conventional wisdom says that the threshold for acceptable fouls is the quarter plus one: i.e. two fouls in the first quarter, three in the second, four in the third, or five in the fourth. As Figure 1 shows, NBA teams are very consistent when applying this rule.

This is a situation that does not apply to a strict interpretation of game theory, as there is only one actor, the coach, who has two choices: sit the star or play the star with foul trouble. Although the observer may believe that wins are the only thing that matters, they are not. The coach must also balance the desire to win with maximizing star player’s times, and keeping the fans happy. These must be balanced in such a way that the team and the coach and the star are all happy.  When handling foul trouble, the decision to bench the star may be consistent with winning the game, but as Weinstein observes, voluntarily benching the star for foul trouble is simply enacting the penalty the coach wishes to avoid.  This is the one situation that depends more on the quality of player (disparity between starter and sub) available to the coach; we will look at some specific examples of NBA teams.

2: Goldman and Rao diagrammed the value of a single point 

in an NBA game relative to the point differential and the time 

remaining. As the game's end nears, the value of a 

single point grows dramatically, especially in 

close games. Source: (Goldman & Rao, 2012)

In 2011, the average number of foul outs per game was 0.3153; I used this number as a proxy to estimate the possibility of a player fouling out given a threshold foul. Goldman and Rao show that the value of a point increases as the game wears on. In order to compensate for this phenomenon I gave quarters 1 and 2 a normal weight, weighted the 3^rd quarter 1.5 times, and gave the fourth quarter 2 times the importance of the first. In order to evaluate the effectiveness of individual players, I used Wins Produced per 48 minutes, available at www.theNBAgeek.com/teams. At any state in the game, there are two particular states, not in threshold foul trouble, or in threshold foul trouble. Once a team’s star player enters the threshold foul trouble state, the coach has two choices. The payoffs of these are shown below for a number of different teams.  This functions as an iterated payoff matrix, where in each quarter, the coach should maximize his expected value.

Expected Value = P_{foul out}(Q_weight)(WP48_player)

Boston Celtics: Paul Pierce vs. Mickael Pietrus

Quarters 1 and 2:

EV_Pierce = (1-0.3153)(1)(0.151)           EV_Pietrus = (0.3153)(1)(0.053)

EV_Pierce = 0.103                                   EV_Pietrus = 0.0167

Quarter 3:

EV_Pierce = (1-0.3153)(1.5)(0.151)        EV_Pietrus = (0.3153)(1.5)(0.053)

EV_Pierce = 0.1545                                 EV_Pietrus = 0.0251

Quarter 4:

EV_Pierce = (0.3153)(2)(0.151)               EV_Pietrus = (1-0.3153)(2)(0.053)

EV_Pierce = 0.2067                                 EV_Pietrus = 0.0334

The fourth quarter is the only time where a player should actually be in a position to foul out; for the rest of the evaluations only the fourth quarter was examined.

Oklahoma City Thunder: Kevin Durant vs. James Harden

EV_Durant = (1-0.3153)(2)(0.226)                      EV_Harden = (0.3153)(2)(0.263)

EV_Durant = 0.3095                                            EV_Harden = 0.1658

Philadelphia 76ers: Andre Iguodala vs. Evan Turner

EV_Iguodala = (1-0.3153)(2)(0.255)                     EV_Turner = (0.3153)(2)(0.111)

EV_Iguodala = 0.3491                                          EV_Turner = 0.0350

Los Angeles Clippers: Chris Paul vs. Mo Williams/ Eric Bledsoe

EV_Iguodala = (1-0.3153)(2)(0.313)                     EV_Turner = (0.3153)(2)((0.024+0.040)/2)

EV_Iguodala = 0.4286                                          EV_Turner = 0.0201

Implication:

3: Since 1987, the number of foul outs per game has 

consistently trended downward. Source: (Feldman, 2010)

Although it is unclear whether foul trouble drives performance or vice versa, it is clear that coaches are being too cautious with their players regarding foul trouble. The benefits of having your star player in the game out-weigh the possible drawbacks of his fouling out. This is especially clear because of the fourth quarter scaling. Although in the first three quarters, the drop off to the bench player may not be as severe, in the fourth quarter, these differences are magnified by the heightened value of a point as the amount of time left in a game nears zero.  This is even true for teams where the disparity in talent between starter and substitute is not drastic. For example, with the Oklahoma City Thunder, should Kevin Durant get into foul trouble in the fourth, the payoff of leaving him in to finish the game is much higher than his value on the bench. It just so happens that, for the Thunder, James Harden’s play has been great enough that the drop off, should Durant foul out, is not particularly problematic.

These results are consistent with findings from Moskowitz and Wertheim (2011), who found that stars actually play better in the fourth quarter with foul trouble. However, this is contrary to Maymin et al., who found that teams generally perform better if foul-troubled starters are benched. Both Maymin et al. and I agree that benching a player in foul trouble is more beneficial in the early quarters, mostly because early in the game, “benching a player preserves “option value” since the coach can reinsert a fresh, non-foul plagued starter back into the game in the fourth quarter” (Maymin, Maymin, & Shen, 2012).

Situation 2: Defend Two or Three Pointer

Often in late game situations, a team may find themselves up by two points with the shot clock turned off. In this situation, the offensive team must decide to go for the tie and hope for overtime, or shoot the three and win the game in regulation. Similarly, the defending team must decide which to defend, the inside or outside shot. For our purposes, it can be assumed that the probability of winning in overtime will be basically a coin-flip, or 50%, as there are too many variables to consider in our theoretical setting (Chow, Miller, Nzima, & Winder, 2012).

In this situation, most often the offensive team’s coach will call a timeout in order to set up his play. Simultaneously, the defensive coach must decide the best way to set his defense in order to ensure the win. This functions as a simultaneous game, where both coaches make their decisions without knowledge of the other’s strategy (Osborne, 2000). Instead of using the data for actual players and teams, I used league averages for my base assumptions regarding the game. According to Weill (2011) tight defense drops expected shooting by 12%. Also, I found data for the effective shooting percentages from Peterson (n.d.) that I feel is representative of the open FG% (although all teams track contested and open FG%, this information is not yet available to the casual fan).

 

League-wide 2 point FG% in 2011-2012: 47.7%

League-wide 3 point FG% in 2011-2012: 34.8%

Open 2 pt. FG% (from eFG%): 62.5% (Peterson, n.d.)

Open 3 pt. FG%: 50.0% (Weill, 2011)

Contested 2 pt. FG%: 35.7% (Weill, 2011)

Contested 3 pt. FG%: 22.8% (Weill, 2011)

The simultaneous game offers no dominant strategy for either team. The teams should therefore employ a mixed strategy in order to remain unpredictable. In order to calculate the right balance of two and three point shots, the Mixed Strategy Equilibrium was calculated for each team.

For offense, let q equal the percentage of time the defending team defends the three. The expected payoff to the shooter is: q * 0.228 + (1 – q) * 0.5 when shooting a three and q * 0.312 + (1 – q) * 0.178 when shooting a two. The offensive team should shoot the three if:

q * 0.228 + (1 – q) * 0.5 > q * 0.312 + (1 – q) * 0.178

This simplifies to q > 0.793, meaning the offensive team should always shoot the three if the defending team defends against the three point shot less than 79.3% of the time. The expected payoff for shooting either a two or a three in this case is 0.284.

For defense, let p equal the percentage of time the offensive team shoots the three. The expected payoff to the defensive team is: p * 0.772 + (1 – p) * 0.688 when defending a three and p * 0.5 + (1 – p) * 0.822 when defending a two. The defensive team should defend the three if:

p * 0.772 + (1 – p) * 0.688 > p * 0.5 + (1 – p) * 0.822

This simplifies to p > 0.330, meaning the defensive team should always defend the three if the offensive team shoots the three point shot more than 33.0% of the time. The expected payoff for defending either a two or a three in this case is 0.716.

Implication: Simply put, it is likely in the best interests of the losing team to shoot the three almost all the time. As long as the defending (winning) team guards the three pointer less than about 80% of the time, the losing team should seek to end the game in regulation every time. Similarly, the team that is ahead should fear the three pointer much more than overtime. As long as the team that is losing shoots the three at least a third of the time, the defending team should always defend the three.

Unfortunately, often finding the best three point shot involves working the ball around and having someone other than the team’s superstar take the shot. In today’s NBA Culture, Hero Ball (Abbott, 2012) has often taken the place of team basketball in crunch time. The problem with this is that isolation plays are good for only 0.78 points per possession (ppp), as opposed to off-the-ball cuts (1.18 ppp) or transition plays (1.12 ppp). When star players do not take the last shot, or when role players miss wide open opportunities, the star is blamed for not taking the shot. However, this analysis shows that the three pointer, especially if the team is able to get off an open look, dramatically improves the team’s chances of winning the game.

Works Cited

Abbott, H. (2012). Hero Ball, or how NBA teams fail by giving the ball to money players in crunch time. ESPN The Magazine. March 19, 2012. Accessed at: http://espn.go.com/nba/story/_/id/7649571/nba-kobe-bryant-not-money-think-espn-magazine

Chow, T., Miller, K., Nzima, S., & Winder, S. (2012). Game Theory (MBA 217) Final Paper. University of California, Berkeley. Accessed at: http://faculty.haas.berkeley.edu/rjmorgan/mba211/Chow%20Heavy%20Industries%20Final%20Project.pdf

Feldman, D. (2010). NBA Players are Fouling Out Less Often. Detroit: PistonPowered.com. Accessed at: http://www.pistonpowered.com/2010/12/nba-players-are-fouling-out-less-often-and-other-interesting-facts-you-didnt-think-you-wanted-to-know-about-fouling-out/

Goldman, M., & Rao, J. M. (2012). Effort vs. Concentration: The Asymmetric Impact of Pressure on NBA Performance. Boston: MIT Sloan Sports Analytics Conference.

Maymin, A., Maymin, P., & Shen, a. E. (2012). How Much Trouble is Early Foul Trouble? Strategically Idling Resources in the NBA. Boston: MIT Sloan Sports Analytics Conference.

Moskowitz, T., & Wertheim, L. (2011). Scorecasting: The Hidden Influences Behind How Sports are Played and Games are Won. New York, NY: Crown Archetype.

Osborne, M. (2000). An Introduction to Game Theory. Oxford: Oxford University Press.

Peterson, E. (n.d.). Open/Contested Shots. 82games.com. Accessed at: http://www.82games.com/saccon.htm

Weill, S. (2011). The Importance of Being Open: What optical tracking data says about NBA field goal shooting. Boston: MIT Sloan Sports Analytics Conference.

Summary Of Findings: Game Theory (4 out of 5 Stars)

Note: This post represents the synthesis of the thoughts, procedures and experiences of others as represented in the 12 articles read in advance of (see previous posts) and the discussion among the students and instructor during the Advanced Analytic Techniques class at Mercyhurst College on 22 APR 2009 regarding Game Theory specifically. This technique was evaluated based on its overall validity, simplicity, flexibility and its ability to effectively use unstructured data.

Description:

Game theory is a method based on applied mathematics and economic theory. It can be useful when attempting to analyze (and ultimately predict) the strategic interactions between two or more actors and the way in which their actions influence future decisions. Game theory assumes that all actors are rational, and can be influenced by various individuals and factors. Games typically involve five common elements: players, strategies, rules, outcomes, and payoffs.

Strengths:

-assumes rational actors

-assumes actors will adjust their actions based on the actions of other actors

-not clearly differentiated from role-playing, simulations, and/or decision trees

-very mathematically based (can be intimidating)

-difficult to quantify options, strategies, and motivations

-may not be a valid method to produce an accurate estimation (see Game Theory, Simulated Interaction, and Unaided Judgment For Forecasting Decisions in Conflict: Further Evidence)

--In real world applications, identifying all of the key players and outcomes can be difficult

Weaknesses:

-Visual step-by-step trail to a conclusion/estimate

-Ability to quantify variables in play

-Emphasis on mathematics and scientifc method

-Applicable to multiple fields (economics, conflict, etc)

-90% rate of success according to BDM

How-To:

Game Theory varies in complexity and in application, however, each application has the following in common:

*Establish the players and the complexity of the game being played, so as to understand the rules which govern the players and the game.

*Identify the possible outcomes for the choices the players can make (although this is particularly difficult as not all decisions can be predicted)

*Establish measurable values for predicted outcomes.

*Eliminate dominated strategies and employ dominate strategies. Repeat this step until a clear, singular strategy emerges or equilibrium is reached between the players.

*Employ selected strategy.

Experience:

As a class, we visited www.gametheory.net and played the repeatable version of Prisoner's Delemma under the "Interactive Materials" tab. Each student played the game at their personal computers. Our objective as we played against the five "personalities" was to identify the particular strategies employed by the computer (in addition to scoring the most utility points). Some of the strategies employed by the computers personalities included "tit-for-tat" and "tit-for-two tats."

Game Theory, Political Economy, and the Evolving Study of War and Peace

By Bruce Bueno de Mesquita

http://www.apsanet.org/imgtest/APSRNov06BuenoDeMesquita.pdf

Summary:

Studies of war and peace increasingly center around domestic interests and institutions for clues on how to shape international affairs. This change in thinking coincides with advances made in non-cooperative game theory and political economy modeling.

De Mesquita refutes realism and state-centric theories as logical explanations for the causes of war. He simultaneously enhances the validity of the liberal peace theory by examining the influences which democrats need to consider when threatening or declaring war.

Realism's claim about a balance of power needed to maintain international stability is refuted by the political economy approach. Simply put, the political economy approach states that the causes of and solutions to international conflict can best be understood by looking within states. It treats leaders as the object of study, not the states as realism does.

A game-theoretic focus concludes that war conducted by rationally acting states is always ex post inefficient. Leaders conduct wars at times to maintain a hold on power, since their domestic constituencies would likely vote them out of office (for democratic states). Autocracies have the advantage of not needing to concern itself with the well-being of their citizenry since they do not face election.

Game theory also validates the liberal peace theory. It claims that leaders (as the object of the political-economy approach to international relations) will only wage war when the outcome is victory, as a 93% success rate for wars initiated by democracies indicate. Since both sides need to consider their reelection prospects, a negotiated settlement to the conflict is the preferred method for resolving conflicts between democratic nations.