1 CHAPTER7 Savvy software agents can encourage the use of second-order theory of mind by negotiators Abstract In social settings, people often rely on their ability to reason about unobservable mental content of other people, such as their beliefs, desires, and intentions. This theory of mind ability helps them to understand and predict the behavior of others. People can even take this ability a step further, and use higher-order theory of mind to reason about the way others use theory of mind to understand and predict behavior, for example by reasoning about the fact Alice believes that Bob does not know about the surprise party. However, empirical evidence suggests that people do not spontaneously use second-order or even higher orders theory of mind in strategic games. In this paper, we let participants negotiate with computational agents of different theory of mind abilities in the setting of Colored Trails. We find that even though participants are unaware of the level of sophistication of their trading partner, within a few rounds of play, participants offers are more indicative of second-order theory of mind reasoning when their computer trading partner was using second-order theory of mind as well. This chapter has previously been published in Proceedings of the 37th Annual Meeting of the Cognitive Science Society (2015). 179
2 Introduction 7.1 Introduction In social settings, people reason about unobservable mental content, such as beliefs, desires, and goals, to predict and interpret the behavior of others. This theory of mind (Premack & Woodruff, 1978) allows people to reason explicitly about the goals of others, such as deciding whether the behavior of others is accidental or intentional. Empirical evidence from second-order false belief tasks (Perner & Wimmer, 1985; Sullivan et al., 1994; Miller, 2009, 2012) reveals that people are also capable of reasoning about the theory of mind of others. People use second-order theory of mind when they reason about the beliefs others have about the beliefs of yet other people, and realize that such nested beliefs can be incorrect. Second-order theory of mind allows people to form nested beliefs such as Alice believes that Bob does not know about the surprise party, and use these beliefs to interpret and predict Alice s behavior. While participants readily use second-order theory of mind reasoning in the second-order false belief task, empirical evidence suggests that in strategic games, participants do not appear to make spontaneous use of higher-order (i.e. at least second-order) theory of mind (Hedden & Zhang, 2002; Camerer et al., 2004; Wright & Leyton-Brown, 2010; Goodie et al., 2012). Over a range of unrepeated single-shot games, Camerer et al. (2004) estimate the distribution of the level of sophistication used by human participants. They find that participant reasoning is typically limited to the use of zero-order or first-order theory of mind. Only few participants are found to be well-described as higher-order theory of mind reasoners (Wright & Leyton-Brown, 2010). When games are repeated, participants can successfully adjust their level of reasoning to accurately predict the behavior of other theory of mind reasoners (Hedden & Zhang, 2002; Zhang et al., 2012; Goodie et al., 2012; Meijering et al., 2010, 2011, 2014; De Weerd et al., 2014b; Devaine, Hollard, & Daunizeau, 2014a), although participants typically need many trials before their behavior is consistent with higher-order reasoning. In this paper, we investigate human-agent interactions in the influential Colored Trails setting, introduced by Grosz, Kraus, and colleagues (Lin et al., 2008; Gal et al., 2010) 1, which provides a useful test-bed to study how different aspects of mixed-motive settings change interactions among agents and humans. In previous work, we presented a computational model for theory of mind agents to study the effectiveness of higher-order theory of mind reasoning in this setting (De Weerd et al., 2013a). We found that the use of first-order and second-order theory of mind allows software agents to balance competitive and cooperative aspects of the game. This way, the use of theory of mind prevents negotiations from breaking down the way they do for agents without theory of mind. In the current paper, we use these agents to determine to what extent human participants reason at 1 Also see
3 7.2. Colored Trails 181 (a) (b) Figure 7.1: In Colored Trails, players spend chips to move around on a 5 by 5 board. (a) To follow the path from location A to location B, a player needs to hand in one blank, one striped, and one dotted chip. (b) Each player is initially located on the central tile S and is assigned a goal location drawn randomly from the gray tiles. higher orders of theory of mind, by letting software agents interact directly with human participants. 7.2 Colored Trails The game we study in this paper is a variation on the influential Colored Trails game. Colored Trails is a board game designed as a research test-bed for investigating decision-making of people and software agents (Lin et al., 2008; Gal et al., 2010). We consider a specific setting in which two negotiating agents alternate in making offers. We have previously used this setting to test the effectiveness of higher-order theory of mind in negotiations (De Weerd et al., 2013a). The game is played on a square board consisting of 25 patterned tiles, like the one depicted in Figure 7.1a. At the start of the game, each player receives a set of four patterned chips, selected at random from the same four possible patterns as those on the board. Each player is initially located on the center tile of the board, indicated with the letter S in Figure 7.1b. The goal of each player is to reach their personal goal location, which is drawn randomly from the board tiles that are at least three steps away from the initial location (gray tiles in Figure 7.1b). To move around on the board, players use their chips. A player can move to a tile adjacent to his current location by handing in a chip of the same pattern as the destination tile. Figure 7.1a shows an example of a Colored Trails board as well as a possible path across the board. A player following the path from location A to the blank tile marked B would have to hand in one striped chip, one dotted chip, and one blank chip. Players are scored based on their success in reaching their goal location. Each
4 Colored Trails player receives 50 points at the start of the game. If the player successfully reaches his goal, he receives an additional 50 points. However, if the goal is not reached, 10 points are deducted from the player s score for each step needed to reach the goal location. Finally, any chip that has not been used to move around the board is worth an additional 5 points to its owner. For example, consider the situation in Figure 7.2, and suppose that player i has goal location G. With his initial set of chips, player i can obtain a score of 50 points. However, if player i would receive one of agent j s blank chips, he could obtain a score of 110 points. To get closer to their goal location, players can trade chips with their coplayer. To capture the dynamic aspect of negotiation, trading among players takes the form of a sequence of alternating offers. When a player makes an offer to redistribute the chips a certain way, his trading partner decides whether or not to accept this offer. If the offer is accepted, the proposed distribution of chips becomes final, the players move as close to their respective goal locations as possible, and the game ends. Alternatively, the trading partner may decide to withdraw from negotiations, which makes the initial distribution final. As a third option, the trading partner may decide to continue the game by rejecting the current offer and making his own offer for a redistribution of chips. There are no restrictions on the offers that players can make. For example, a player is allowed to repeat an offer that has been previously rejected by his trading partner, or make an offer that he has previously rejected himself. However, the game ends if six offers have been rejected. In any game, each player can therefore make at most three offers. If a game ends because the maximum number of offers has been exceeded, the initial distribution of chips becomes final. Although a player s score is based only on how closely he approaches his own goal, Colored Trails is not a purely competitive game. Since a player may need a different set of chips to achieve his goal than his trading partner, there may be an opportunity for a cooperative trade that allows both players to obtain a higher score. That is, although the score of a player does not depend on how closely his trading partner approaches his goal location, players can still benefit from taking into account the goal of their trading partner. Importantly, however, Colored Trails is a game of imperfect information: while players know what chips are in possession of their trading partner, they do not know the goal location of their trading partner at the start of the game. In this paper, we investigate to what extent human participants reason using higher orders of theory of mind when playing Colored Trails with a software agent as trading partner, and to what extent participants adjust their level of theory of mind reasoning in response to the behavior of their trading partner. Since simulation experiments with agents have shown that second-order theory of mind can help agents to avoid negotiation failure and balance cooperative and competitive aspects of the game (De Weerd et al., 2013a), the Colored Trails setting may facilitate theory of mind reasoning in human participants.
5 7.3. Theory of mind software agents Theory of mind software agents The theory of mind agents presented here as trading partners of human participants are adapted from De Weerd et al. (2013a) to allow for games with a known finite horizon. That is, the computational agents know that the game cannot last more than six turns. In this section, we describe the way these make use of theory of mind. The mathematical details of these agents can be found in De Weerd et al. (2013a) Zero-order theory of mind A zero-order theory of mind (ToM 0 ) agent is unable to reason about unobservable mental content of its trading partner, including its goal location. Instead, a ToM 0 agent models the behavior of its trading partner in terms of the offers that the trading partner is willing to accept. Based on previous experience in the Colored Trails game, a ToM 0 agent constructs beliefs about the likelihood that certain offers will be accepted by the trading partner. For example, over repeated games, a ToM 0 agent will learn that the trading partner rarely accepts offers that assign few chips to the trading partner, while offers that assign many chips to the trading partner are accepted with a high frequency. Using these zero-order beliefs, a ToM 0 agent can calculate the expected gain of making a particular offer, and choose the action that the agent expects to yield it the highest gain. Based on the actions of the trading partner, the ToM 0 agent then updates its zero-order beliefs. This way, the ToM 0 agent can play Colored Trails without attributing any mental content to others. In particular, although the ToM 0 agent s zero-order beliefs eventually reflect the desires of its trading partner, the ToM 0 agents cannot reason about such desires explicitly. In terms of negotiation strategies, the ToM 0 agent engages purely in positional bargaining (Fisher & Ury, 1981), by reasoning only about offers and the likelihood that these offers will be accepted by its trading partner First-order theory of mind In addition to its zero-order beliefs, a first-order theory of mind ToM 1 agent can also determine what its own decision would have been if it had been in the position of its trading partner. This way, a ToM 1 agent can consider that its trading partner has beliefs and goals similar to its own that determine whether or not an offer will be accepted. A ToM 1 agent believes that an offer will only be accepted if it increases the score of both the agent itself and its trading partner since the ToM 1 agent itself would only accept offers that increase its own score. In the same way, the ToM 1 agent realizes that its trading partner only makes offers that would increase its own
6 Theory of mind software agents Figure 7.2: Example of a negotiation setting in Colored Trails. Agent j offers to trade the striped chip owned by agent i against the dotted chip owned by agent j. Since this trade would make it harder for agent i to reach his goal location (tile G), agent i will reject this offer. The goal location of agent j is not shown. score. This means that the offers made by the trading partner contain information about its goal location. For example, consider the situation depicted in Figure 7.2. In this example, agent j offers to trade its dotted chip for the striped chip owned by agent i. From this offer, a ToM 1 agent would conclude that the striped chip allows agent j to move closer to its goal location. Secondly, since the offered trade would leave agent j without any dotted chips, a ToM 1 agent would believe that agent j does not need any dotted chips to reach its goal location. This excludes several possible goal locations for agent j. Importantly, a ToM 1 agent s first-order theory of mind is additional to its zero-order beliefs. Through repeated interactions, a ToM 1 agent may come to believe that first-order theory of mind fails to accurately model the behavior of its trading partner and that the use of zero-order theory of mind would result in a higher score. In this case, a ToM 1 agent may decide to play Colored Trails as if he were a ToM 0 agent Second-order theory of mind Agents capable of second-order theory of mind can also consider the possibility that their trading partner is a ToM 1 agent. A second-order theory of mind (ToM 2 ) agent believes that its trading partner may be trying to interpret the offers made by the ToM 2 agent to determine the ToM 2 agent s goal. This allows the ToM 2 agent to reason about the way different offers influence the beliefs of the trading partner about the agent s goal, and select the offer that provides its trading partner with as much information about its goal location as possible. For example, suppose agent i in Figure 7.2 is a ToM 2 agent with goal location G. Using second-order theory of mind, agent i knows that making any offer in which the striped chip is assigned to agent j, agent j can conclude that agent i does not need a striped chip to reach its goal location. This allows agent j to
7 7.4. Methods 185 exclude many possible goal locations for agent i, which can help agent j to make an offer that is acceptable to agent i. In this case, although agent i knows that agent j has a goal location, agent i remains unaware of what that goal location is. A second-order theory of mind agent can therefore engage in interest-based negotiation (Fisher & Ury, 1981), by choosing its offers in such a way that they communicate the agent s interests to its trading partner. Similar to the ToM 1 agent, a ToM 2 agent does not know the extent of its trading partner s theory of mind abilities. Instead, a ToM 2 agent has zero-order, first-order, and second-order beliefs about the behavior of its trading partner. While negotiating in Colored Trails, the ToM 2 agent keeps updating its beliefs concerning which of these beliefs most accurately describes the actual behavior of its trading partner. This means that a ToM 2 agent may sometimes behave as if it were a ToM 0 agent, while behaving like a ToM 2 agent on other occasions. 7.4 Methods Participants Twenty-seven students (10 female) of the University of Groningen participated in this study. All participants were informed that after the conclusion of the study, the three participants with the highest score in the negotiation game received e15, e10, and e5, respectively. Each participant gave informed consent prior to admission into the study Materials Twenty-four games were selected from a set of randomly generated games. To ensure that these games would allow us to distinguish between different orders of theory of mind reasoning of participants, they were selected so that: The participant s goal could be reached with the eight chips in the game; Simulations with computational agents predicted different outcomes for participants using zero-order, first-order, and second-order theory of mind; and Simulations with computational agents predicted that the game would last at least two turns and at most six turns. These games were divided into three blocks of eight games each. The level of theory of mind reasoning of the software agent was fixed within each block, and varied between blocks.
8 Methods Design and procedure Before the start of the experiment, participants were tested on colorblindness. Participants were asked to distinguish patches of blue and orange, with four possible intensities of each color. All participants passed the colorblindness test. Next, participants played several Marble Drop games (Meijering et al., 2011). The Colored Trails experiment consisted of a familiarization phase and an experimental phase. At the start of the familiarization phase, participants were asked to imagine themselves as an attorney for a major corporation. In this function, they would be involved in a number of negotiations with different clients. Participants were told that their trading partner was a computer player (Alex), which would always react on their offer as quickly as possible in a way it believed would maximize its own score. To ensure understanding of the Colored Trails game, participants answered a few questions about the rules, scoring, and movements on the game board. In the experimental phase, participants played three blocks of eight games each. In each block, the participant either faced a ToM 0, ToM 1, or ToM 2 agent. The order of the blocks was randomized across participants. Participants were not informed that the level of reasoning of the trading partner would change over the course of the experiment, but participants were told that they would face different clients. At the start of the experiment, it was randomly decided whether the participant or the software agent would make the initial offer of the first game. In subsequent games, participant and agent alternated in the role of initiating player. Participants were allowed 60 seconds to decide on their next action. During each round, the remaining decision time was presented to participants by means of a countdown timer. If a participant had not made a decision within 60 seconds, the game continued without an offer being made, and the software agent took its turn. The zero-order beliefs of theory of mind agents were initialized by playing 200 randomly generated Colored Trails games against another agent. This allowed agents to learn what kind of offers were more likely to be acceptable to their trading partner. To conform to our cover story in which participants were told that they would face a number of different clients, the agent s beliefs were reset to this initial value at the start of each game. Additionally, theory of mind agents started every game reasoning at the highest order of theory of mind available to them. This means that although software agents learned from a participant s offers within a single game, and adjusted their behavior accordingly, agents did not exhibit any learning across games. This way, agents were prevented from adapting to specific participants, and every participant faced the same agent in every scenario. After the Colored Trails games, participants answered a short questionnaire
9 7.5. Results 187 Agent's order of theory of mind Zero order First order Second order Agent performance 0 0 Participant performance Figure 7.3: Outcomes of Colored Trails per block. The solid line shows the Pareto optimal outcomes. Dashed lines show the score participants and software agents would achieve if they were to withdraw from negotiation in every game. about the perceived difficulty of the task, the behavior of their trading partner, and the participant s reasoning strategies. In addition, participants took a test for their interpersonal reactivity index (Davis, 1983). 7.5 Results Figure 7.3 shows the outcomes of the Colored Trails game. The graph shows how the score of agents and participants changed as a result of negotiation for each participant and for each block. Dashed lines indicate the zero performance line, which is the score that players would have received if every game of the block had ended with withdrawal from negotiation. A score below the dashed line indicates that a player decreased its score through negotiation. As Figure 7.3 shows, only once a participant received a negative score in one of the blocks. The solid line in Figure 7.3 shows the boundary of Pareto efficient outcomes. This boundary shows those outcomes for which neither the participant nor the software agent could have received a higher score without a decrease in the score of the other player. The Pareto boundary gives an impression of how well partici-
10 Results Order of theory of mind used by participant Zero order First order Second order Confidence in orders of theory of mind Zero order First order Second order Agent's order of theory of mind Figure 7.4: Estimated similarity of participant offers to the offers of ToM 0 (red circles), ToM 1 (green triangles), and ToM 2 (blue squares) agents in each of the three blocks. Brackets indicate standard error. pants and software agents played Colored Trails. Participants and software agents generally negotiated mutually beneficial solutions, while neither player systematically exploited the other. Additionally, Figure 7.3 shows that when participants negotiate with ToM 2 agents, they tend to end up closer to the Pareto optimality, while negotiations between participants and ToM 1 agents typically end up far from the Pareto boundary. Importantly, the use of theory of mind agents allows us to estimate to what extent participants make use of theory of mind while playing Colored Trails. We use a ToM 3 spectator agent that observes the offers of a participant and determines whether these offers are most consistent with zero-order, first-order, or second-order theory of mind reasoning. The software agent constructs a confidence for each order of theory of mind at which it can reason to decide which order of theory of mind would yield the best outcome (De Weerd et al., 2013a). Each time the participant makes an offer O, the ToM 3 agent updates its confidence that this participant is using kth-order theory of mind by calculating the likelihood that a ToM k agent would have made an offer similar to offer O. For each of the three blocks, Figure 7.4 shows how similar participant offers were to offers of ToM 0, ToM 1, and ToM 2 agents, as judged by the ToM 3 agent.
11 7.6. Discussion and conclusion 189 Red circles indicate the average similarity of a participant s offers to zero-order theory of mind reasoning, green triangles indicate the similarity to first-order theory of mind reasoning, and blue squares show the similarity to second-order theory of mind reasoning. Interestingly, Figure 7.4 shows that participant offers are more similar to first-order and second-order theory of mind reasoning than they are to zero-order theory of mind reasoning. Figure 7.4 also shows that the similarity ratings of participant offers vary depending on the order of theory of mind of the computer trading partner. Although similarity ratings for zero-order and first-order theory of mind reasoning show no variation across different levels of sophistication of the trading partner (X(2) 2 = 0.52, ns, and X2 (2) = 2.67, ns, respectively), participant offers were significantly more similar to second-order theory of mind reasoning when they were facing a ToM 2 trading partner (X(2) 2 = 24.89, p < 0.001). Previous studies into negotiations show that the opening bid of a negotiation can serve as an anchor for the entire negotiation process, making the first bid of a game especially influential in the negotiation process (Raiffa et al., 2002; Van Poucke & Buelens, 2002). In our experiment, the identity of the initiating player indeed influences negotiation outcomes. In general, both players ended up with an extra 15 points on average after negotiation when the software agent made the initial offer rather than when the participant was the first to propose a trade. The only exception to this rule was that participants negotiating with a ToM 2 agent ended up with a higher score when they made the initial offer themselves. This effect can be explained by the way agents of different orders of theory of mind construct their offers. Both ToM 0 and ToM 1 agents make offers that they believe will be accepted by their trading partner. In contrast, ToM 2 agents make offers that inform their trading partner about their own goals. As a result, initial offers made by ToM 0 and ToM 1 agents are typically more favorable to their trading partner than those made by ToM 2 agents. Similarly, when participants reasoned more like ToM 2 agents, their initial offers were more favorable to themselves than to their trading partner. 7.6 Discussion and conclusion Experimental evidence suggests that participants do not make spontaneous use of higher-order theory of mind reasoning in unrepeated games (Hedden & Zhang, 2002; Camerer et al., 2004; Wright & Leyton-Brown, 2010; Goodie et al., 2012), though participants can successfully adjust their level of reasoning to accurately predict the behavior of other theory of mind reasoners (Hedden & Zhang, 2002; Goodie et al., 2012; Meijering et al., 2010, 2011, 2014; Devaine et al., 2014a). Surprisingly, in this paper, we find that participants show behavior consistent
12 Discussion and conclusion with second-order theory of mind reasoning in a negotiation game that lasts a few rounds only. In our experiments, human participants negotiated with software agents that dynamically change their order of theory of mind reasoning in response to the behavior of their trading partner. We use these agent-based models to analyze participant behavior in a dynamic setting. Our model explicitly takes into account that participants may differ in the order of theory of mind at which they reason, and that a participant may change the order of theory of mind at which he reasons over the course of a single game. Based on this agent-based analysis, we find that participants make offers that are more consistent with second-order theory of mind reasoning when their trading partner is capable of second-order theory of mind as well. Interestingly, while participants knew that they would face different trading partners, they were unaware that these trading partners differed in their theory of mind abilities. That is, the behavior of higher-order theory of mind agents apparently encouraged participants to make use of higher-order theory of mind as well. Experiments with adults typically show that individuals reason at low orders of theory of mind, and are slow to adjust to an opponent that reasons using theory of mind (Hedden & Zhang, 2002; Camerer et al., 2004; Wright & Leyton-Brown, 2010; Goodie et al., 2012). In our setting, however, participants exhibited secondorder theory of mind within a few games. It is possible that the negotiation setting, which involves both cooperative and competitive goals, emphasized the social nature of the task. Such social framing has been shown to encourage the use of theory of mind (Goodie et al., 2012; Devaine et al., 2014a). Our results show that mixed groups of human and software agents can successfully negotiate a mutually beneficial outcome. However, none of the negotiation outcomes in our experiment were Pareto efficient. That is, each participant could have received a higher score without reducing the score of their trading partner. This indicates that there is still room for significant improvement. One factor that may have limited the effectiveness of negotiations in Colored Trails is the time limit on the decisions of participants. Although participants failed to make a decision before time ran out in only four occasions, it is likely that participants selected suboptimal actions due to time constraints. Removing this time constraint in future experiments may increase negotiation performance. Our results suggest that computational theory of mind agents can be used as a training tool for negotiation. When participants negotiated with a trading partner capable of second-order theory of mind, the outcome was generally closer to a Pareto optimal solution than when participants faced less sophisticated trading partners. In addition, our results show that agents could benefit more from making the opening bid than participants. Experience with theory of mind agents may therefore allow participants to learn to leverage the anchoring effect of the initial offer.
13 7.7. Acknowledgments Acknowledgments This work was supported by the Netherlands Organisation for Scientific Research (NWO) Vici grant NWO , awarded to Rineke Verbrugge for the project Cognitive systems in interaction: Logical and computational models of higherorder social cognition.
Savvy software agents can encourage the use of second-order theory of mind by negotiators Harmen de Weerd, Eveline Broers, Rineke Verbrugge Institute of Artificial Intelligence, University of Groningen
This space is reserved for the Procedia header, do not use it Higher-order theory of mind in Tacit Communication Game Harmen de Weerd 1, Rineke Verbrugge 1, and Bart Verheij 1,2 1 Institute of Artificial
The Role of Simple and Complex Working Memory Strategies in the Development of First-order False Belief Reasoning: A Computational Model of Transfer of Skills Burcu Arslan (email@example.com), Stefan
Metacognition and Mathematical Problem Solving: Helping Students to Ask The Right Questions Shirley Gartmann and Melissa Freiberg The acquisition of problem solving, reasoning and critical thinking skills
Gaming the Law of Large Numbers Thomas Hoffman and Bart Snapp July 3, 2012 Many of us view mathematics as a rich and wonderfully elaborate game. In turn, games can be used to illustrate mathematical ideas.
Section 6.2 Definition of Probability Probability is a measure of the likelihood that an event occurs. For example, if there is a 20% chance of rain tomorrow, that means that the probability that it will
The Learning Skills Pyramid Brett A. Brosseit, 2013 To develop strong critical thinking and legal analysis skills, students need to: Develop new patterns of thinking Understand the mental processes they
Introduction to Hypothesis Testing CHAPTER 8 LEARNING OBJECTIVES After reading this chapter, you should be able to: 1 Identify the four steps of hypothesis testing. 2 Define null hypothesis, alternative
Why is Insurance Good? An Example Jon Bakija, Williams College (Revised October 2013) Introduction The United States government is, to a rough approximation, an insurance company with an army. 1 That is
First and second-order false-belief reasoning: Does language support reasoning about the beliefs of others? Bart Hollebrandse, Angeliek van Hout and Petra Hendriks University of Groningen 1 Introduction
CHANCE ENCOUNTERS Making Sense of Hypothesis Tests Howard Fincher Learning Development Tutor Upgrade Study Advice Service Oxford Brookes University Howard Fincher 2008 PREFACE This guide has a restricted
MAKING MATH MORE FUN BRINGS YOU FUN MATH GAME PRINTABLES FOR HOME OR SCHOOL THESE FUN MATH GAME PRINTABLES are brought to you with compliments from Making Math More Fun at and Math Board Games at Copyright
Ion Propulsion Ion Propulsion Engine Simulation STUDENT ACTIVITY AND REPORT SHEET This activity must be completed at a computer with Internet access. Part 1: Procedure 1. Go to http://dawn.jpl.nasa.gov/mission/ion_engine_interactive/index.html
Washington State K 8 Mathematics Standards Data Analysis, Statistics, and Probability Strand In kindergarten through grade 5, students learn a variety of ways to display data, and they interpret data to
Agent Simulation of Hull s Drive Theory Nick Schmansky Department of Cognitive and Neural Systems Boston University March 7, 4 Abstract A computer simulation was conducted of an agent attempting to survive
Privacy/Efficiency Tradeoffs in Distributed Meeting Scheduling by Constraint Based Agents 1 Eugene C. Freuder, Marius Minca and Richard J. Wallace Constraint Computation Center Department of Computer Science,
Problem 3 If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C? Suggested Questions to ask students about Problem 3 The key to this question
Research Report Banking on a Bad Bet: Probability Matching in Risky Choice Is Linked to Expectation Generation Psychological Science 22(6) 77 711 The Author(s) 11 Reprints and permission: sagepub.com/journalspermissions.nav
Ignoring base rates People were told that they would be reading descriptions of a group that had 30 engineers and 70 lawyers. People had to judge whether each description was of an engineer or a lawyer.
STA 371G: Statistics and Modeling Decision Making Under Uncertainty: Probability, Betting Odds and Bayes Theorem Mingyuan Zhou McCombs School of Business The University of Texas at Austin http://mingyuanzhou.github.io/sta371g
Milgram Activities Activity 1 Procedures Aim: To consolidate students knowledge of the procedures of Milgram s study and to encourage co-operative working amongst students. Cut up the following table of
Week 7 - Game Theory and Industrial Organisation The Cournot and Bertrand models are the two basic templates for models of oligopoly; industry structures with a small number of firms. There are a number
The Economic Value of Certainty By Les McGuire, MBA August 19, 2003 This is supposed to be an article about Whole Life Insurance, and for those who have eyes to see it is. However, before discussing product,
SPERNER S LEMMA AND BROUWER S FIXED POINT THEOREM ALEX WRIGHT 1. Intoduction A fixed point of a function f from a set X into itself is a point x 0 satisfying f(x 0 ) = x 0. Theorems which establish the
A Client-Server Interactive Tool for Integrated Artificial Intelligence Curriculum Diane J. Cook and Lawrence B. Holder Department of Computer Science and Engineering Box 19015 University of Texas at Arlington
How to Construct a Believable Opponent using Cognitive Modeling in the Game of Set Niels A. Taatgen (firstname.lastname@example.org) Marcia van Oploo (email@example.com) Jos Braaksma (firstname.lastname@example.org) Jelle Niemantsverdriet
Teacher s Guide Getting Started Shereen Khan & Fayad Ali Trinidad and Tobago Purpose In this two-day lesson, students develop different strategies to play a game in order to win. In particular, they will
Five Myths of Active Portfolio Management Most active managers are skilled. Jonathan B. Berk Proponents of efficient markets argue that it is impossible to beat the market consistently. In support of their
Curriculum Map Statistics and Probability Honors (348) Saugus High School Saugus Public Schools 2009-2010 Week 1 Week 2 14.0 Students organize and describe distributions of data by using a number of different
Volume of Pyramids and Cones Objective To provide experiences with investigating the relationships between the volumes of geometric solids. www.everydaymathonline.com epresentations etoolkit Algorithms
The Partial-Quotients Division Algorithm, Part 1 Objectives To introduce and provide practice with a low-stress division algorithm for 1-digit divisors. www.everydaymathonline.com epresentations etoolkit
Balanced Equations Current California Math Standards Balanced Equations Grade Three Number Sense 1.0 Students understand the place value of whole numbers: 1.1 Count, read, and write whole numbers to 10,000.
Introduction to time series analysis Margherita Gerolimetto November 3, 2010 1 What is a time series? A time series is a collection of observations ordered following a parameter that for us is time. Examples
www.sciencemag.org/cgi/content/full/319/5862/414/dc1 Supporting Online Material for Application of Bloom s Taxonomy Debunks the MCAT Myth Alex Y. Zheng, Janessa K. Lawhorn, Thomas Lumley, Scott Freeman*
Cryptography and Network Security Prof. D. Mukhopadhyay Department of Computer Science and Engineering Indian Institute of Technology, Karagpur Lecture No. #06 Cryptanalysis of Classical Ciphers (Refer
374 Chapter 8 The Mathematics of Likelihood 8.3 Expected Value Find an expected value involving two events. Find an expected value involving multiple events. Use expected value to make investment decisions.
Random Fibonacci-type Sequences in Online Gambling Adam Biello, CJ Cacciatore, Logan Thomas Department of Mathematics CSUMS Advisor: Alfa Heryudono Department of Mathematics University of Massachusetts
A Concrete Introduction to the Abstract Concepts of Integers and Algebra using Algebra Tiles Table of Contents Introduction... 1 page Integers 1: Introduction to Integers... 3 2: Working with Algebra Tiles...
Using Puzzles to Teach Problem Solving TEACHER S GUIDE TO IZZI AND IZZI 2 BENEFITS IZZI and IZZI 2 are a pattern matching games that are an exciting way to teach elements of problem solving pattern recognition
A wargame with dynamic pieces for two to six players. by Jonathan A. Leistiko Object Drive your opponents out of the game by running them out of resources or invading their home base. You Need A MetalTalon
Ground Truthing Policy Using Participatory Map-Making to Connect Citizens and Decision Makers Shalini P. Vajjhala Citizen participation has become an increasingly important component of development planning
The foreign King Introduction Incited by the economic and political interests of the bourgeoisie a liberal regime of monarchy was constituted in Belgium on 1830 gaining the independence from the United
Introduction to Applied Behavior Analysis For a variety of reasons, behavior patterns develop differently in children, and the development can be more variable in children with disabilities like autism
Algorithms and optimization for search engine marketing Using portfolio optimization to achieve optimal performance of a search campaign and better forecast ROI Contents 1: The portfolio approach 3: Why
Vern Lindberg 1 Making Graphs A picture is worth a thousand words. Graphical presentation of data is a vital tool in the sciences and engineering. Good graphs convey a great deal of information and can
2DI36 Statistics 2DI36 Part II (Chapter 7 of MR) What Have we Done so Far? Last time we introduced the concept of a dataset and seen how we can represent it in various ways But, how did this dataset came
An Investigation into Visualization and Verbalization Learning Preferences in the Online Environment Dr. David Seiler, Assistant Professor, Department of Adult and Career Education, Valdosta State University,
AP Stats - Probability Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. I toss a penny and observe whether it lands heads up or tails up. Suppose
Integrated Water Resources Management (Proceedings of a symposium held al Davis. California. April 2000). IAHS Publ. no. 272. 2001. 37 Collaborative planning in integrated water resources management: the
Basic Concepts in Research and Data Analysis Introduction: A Common Language for Researchers...2 Steps to Follow When Conducting Research...3 The Research Question... 3 The Hypothesis... 4 Defining the
Moral Hazard Itay Goldstein Wharton School, University of Pennsylvania 1 Principal-Agent Problem Basic problem in corporate finance: separation of ownership and control: o The owners of the firm are typically
Expected Value 24 February 2014 Expected Value 24 February 2014 1/19 This week we discuss the notion of expected value and how it applies to probability situations, including the various New Mexico Lottery
Game Theory Themes 1. Introduction to Game Theory 2. Sequential Games 3. Simultaneous Games 4. Conclusion Introduction to Game Theory Game theory is the branch of decision theory concerned with interdependent
Age to Age Factor Selection under Changing Development Chris G. Gross, ACAS, MAAA Introduction A common question faced by many actuaries when selecting loss development factors is whether to base the selected
SIX-STEP PROBLEM SOLVING MODEL Problem solving models are used to address many issues that come up on a daily basis in the workplace. These problems may be technical or issue-based. While many of you have
Quarterly Journal of Experimental Psychology (1971) 23, 63-71 NATURAL AND CONTRIVED EXPERIENCE IN A REASONING PROBLEM P. C. WASON AND DIANA SHAPIRO Psycholinguistics Research Unit, Department of Phonetics,
The Collaborative Service Delivery Matrix: A Decision Tool to Assist Local Governments A Product of the Enhanced Partnership of the ICMA, the Alliance for Innovation, and the Center for Urban Innovation
PoW-TER Problem Packet A Phone-y Deal? (Author: Peggy McCloskey) 1. The Problem: A Phone-y Deal? [Problem #3280] With cell phones being so common these days, the phone companies are all competing to earn
Lab 2.1 Tracking Down the Bugs Chapter 7 (To Err is Human ) discusses strategies for debugging finding and fixing problems with IT systems. In this lab, we focus on the early stages of debugging, where
Abstract Title Page. Title: Conditions for the Effectiveness of a Tablet-Based Algebra Program Authors and Affiliations: Andrew P. Jaciw Megan Toby Boya Ma Empirical Education Inc. SREE Fall 2012 Conference
Key Terms for This Session Session 8 Probability Previously Introduced frequency New in This Session binomial experiment binomial probability model experimental probability mathematical probability outcome
Michael Lacewing Descartes rationalism Descartes Meditations provide an extended study in establishing knowledge through rational intuition and deduction. We focus in this handout on three central claims:
15-251: Great Theoretical Ideas in Computer Science Anupam Gupta Notes on Combinatorial Games (draft!!) January 29, 2012 1 A Take-Away Game Consider the following game: there are 21 chips on the table.
Math 728 Lesson Plan Tatsiana Maskalevich January 27, 2011 Topic: Probability involving sampling without replacement and dependent trials. Grade Level: 8-12 Objective: Compute the probability of winning
36 Odds, Expected Value, and Conditional Probability What s the difference between probabilities and odds? To answer this question, let s consider a game that involves rolling a die. If one gets the face
Paper presented at the Third International Conference on Complex Systems, Nashua, NH, April, 2000. Please do not quote from without permission. Agent-based Modeling of Disrupted Market Ecologies: A Strategic
Groundbreaking Technology Redefines Spam Prevention Analysis of a New High-Accuracy Method for Catching Spam October 2007 Introduction Today, numerous companies offer anti-spam solutions. Most techniques
All Policies Are Glocal: International Environmental Policymaking with Strategic Subnational Governments Online Appendix Michael M. Bechtel Johannes Urpelainen June 12, 2014 This appendix provides more
Uses of Java Applets in Mathematics Education Christopher P. Mawata Mathematics Department University of Tennessee at Chattanooga email@example.com Abstract: This paper illustrates different ways
InterNeg Teaching Notes 1/11 E procurement: Multiattribute Auctions and Negotiations Gregory E. Kersten InterNeg Research Center and J. Molson School of Business, Concordia University, Montreal 1. Introduction
University of Oslo Department of Economics Exam: ECON3200/4200 Microeconomics and game theory Date of exam: Tuesday, November 26, 2013 Grades are given: December 17, 2013 Duration: 14:30-17:30 The problem
Chapter 1 Assignment Part 1 Careers in Psychology 1. Which of the following psychological professionals must always have a medical degree? a. psychologist b. psychiatric social worker c. psychiatrist d.
Course: Introduction to Sociology Kirkwood Community College CASTLE Case Study Analysis Questions Introduction to Sociology Jeff Sherman May 4, 2004 Learning Outcomes: Evaluate the techniques of different
Research Article P vs NP problem in the field anthropology Michael.A. Popov, Oxford, UK Email Michael282.firstname.lastname@example.org Keywords P =?NP - complexity anthropology - M -decision - quantum -like game - game-theoretical
Sam Chow One Player Games Bored MUMSians Han writes the numbers 1, 2,..., 100 on the whiteboard. He picks two of these numbers, erases them, and replaces them with their difference (so at some point in
A Probability puzzler!! Basic Probability Theory I Dr. Tom Ilvento FREC 408 Our Strategy with Probability Generally, we want to get to an inference from a sample to a population. In this case the population
How to Conduct a Hypothesis Test The idea of hypothesis testing is relatively straightforward. In various studies we observe certain events. We must ask, is the event due to chance alone, or is there some
Which Design Is Best? Which Design Is Best? In Investigation 2-8: Which Design Is Best? students will become more familiar with the four basic epidemiologic study designs, learn to identify several strengths
Table of Contents Executive Summary 3 Introduction 4 Who manages the RFP process? 4 The Request for Proposal Process 4 Stage One: Contractor identifies and documents service delivery 5 Stage Two: Buyer