Game theory is the study of mathematical models of conflict and cooperation between rational decision-makers. It analyzes strategic decision-making through modeling games with several players under conditions of both cooperation and conflict. Game theory looks at solution concepts such as Nash equilibria, which are strategy profiles where each player's strategy is a best response to the other players' strategies. It is used to understand outcomes in strategic interactions in economics, political science, and other fields.
This document discusses the costs of taxation in terms of deadweight loss. It explains that taxes reduce overall welfare by creating a wedge between the price paid by buyers and received by sellers. This leads to a reduction in quantity traded below the efficient market level. The deadweight loss is the loss of overall surplus, and grows as the tax increases and distorts market incentives more. Tax revenue initially rises with the tax rate but eventually falls as the tax starts to significantly reduce the size of the market. The deadweight loss from taxation depends on the price elasticities of supply and demand.
This document provides an overview of tariffs as an instrument of trade policy. It discusses:
1) The objectives of understanding tariffs and their effects on trade patterns, welfare, and income distribution.
2) Models used to analyze the effects of tariffs, including partial equilibrium models and examining small vs. large country cases.
3) How tariffs affect prices, consumption, production, trade balances, and welfare in importing and exporting countries through a shift in supply and demand curves. Tariffs create costs through deadweight losses.
4) Concepts of consumer surplus, producer surplus, and total surplus are used to measure and compare the costs and benefits of tariffs, free trade, and autark
This document provides definitions of economics from different perspectives and outlines the basic concepts and principles of managerial economics. It discusses how economics can be viewed as both a science and an art. Microeconomics studies individual actors like firms and households while macroeconomics looks at aggregates. Managerial economics applies economic theory to business decision making under uncertainty. It helps address resource allocation, inventory, pricing, and investment problems. Managerial economics is related to other fields like operations research, decision theory, statistics, and accounting.
1) The document summarizes key concepts from Chapter 3 of a macroeconomics textbook, including how total national income is determined by aggregate supply and demand.
2) It explains how factor prices like wages and rental rates are determined by supply and demand in factor markets and how this determines the distribution of total income.
3) It outlines the components of aggregate demand - consumption, investment, and government spending - and how their interaction with aggregate supply determines equilibrium in the goods market and the loanable funds market.
What is Demand?
Diff. bet Demand and quantity demand
Types of demand - Individual and Market
What is the Law of Demand?
Assumptions of Law of Demand
Why demand curve sloping downward?
Reasons for inverse relationship
Determinents of Demand
What is Band Wagon & Snob effect
Deadweight loss refers to the decrease in total surplus that results from an inefficient allocation of resources due to market failures like price and quantity restrictions, taxes and subsidies, monopoly power, or externalities. These market inefficiencies can lead to either underproduction or overproduction. Price ceilings and floors can create deadweight loss by causing excess supply or demand when the price is set above or below the market clearing price. Ignoring externalities, like pollution, can also lead to inefficient outcomes if social costs and benefits are not accounted for in supply and demand decisions. Monopolies restrict supply and raise prices above competitive levels, creating another source of deadweight loss.
This document summarizes key concepts from Chapter 4 of an International Economics textbook. It covers theories for trade protection such as the infant industry argument. It then discusses tariffs, including import/export tariffs and calculations of effective rates of protection. Finally, it examines nontariff barriers such as import quotas, tariff-rate quotas, and subsidies. For quotas and tariffs, it outlines the impacts on consumer surplus, producer surplus, and government revenue. The chapter suggests quotas impose larger losses than equivalent tariffs due to restrictions on consumption.
This document discusses the costs of taxation in terms of deadweight loss. It explains that taxes reduce overall welfare by creating a wedge between the price paid by buyers and received by sellers. This leads to a reduction in quantity traded below the efficient market level. The deadweight loss is the loss of overall surplus, and grows as the tax increases and distorts market incentives more. Tax revenue initially rises with the tax rate but eventually falls as the tax starts to significantly reduce the size of the market. The deadweight loss from taxation depends on the price elasticities of supply and demand.
This document provides an overview of tariffs as an instrument of trade policy. It discusses:
1) The objectives of understanding tariffs and their effects on trade patterns, welfare, and income distribution.
2) Models used to analyze the effects of tariffs, including partial equilibrium models and examining small vs. large country cases.
3) How tariffs affect prices, consumption, production, trade balances, and welfare in importing and exporting countries through a shift in supply and demand curves. Tariffs create costs through deadweight losses.
4) Concepts of consumer surplus, producer surplus, and total surplus are used to measure and compare the costs and benefits of tariffs, free trade, and autark
This document provides definitions of economics from different perspectives and outlines the basic concepts and principles of managerial economics. It discusses how economics can be viewed as both a science and an art. Microeconomics studies individual actors like firms and households while macroeconomics looks at aggregates. Managerial economics applies economic theory to business decision making under uncertainty. It helps address resource allocation, inventory, pricing, and investment problems. Managerial economics is related to other fields like operations research, decision theory, statistics, and accounting.
1) The document summarizes key concepts from Chapter 3 of a macroeconomics textbook, including how total national income is determined by aggregate supply and demand.
2) It explains how factor prices like wages and rental rates are determined by supply and demand in factor markets and how this determines the distribution of total income.
3) It outlines the components of aggregate demand - consumption, investment, and government spending - and how their interaction with aggregate supply determines equilibrium in the goods market and the loanable funds market.
What is Demand?
Diff. bet Demand and quantity demand
Types of demand - Individual and Market
What is the Law of Demand?
Assumptions of Law of Demand
Why demand curve sloping downward?
Reasons for inverse relationship
Determinents of Demand
What is Band Wagon & Snob effect
Deadweight loss refers to the decrease in total surplus that results from an inefficient allocation of resources due to market failures like price and quantity restrictions, taxes and subsidies, monopoly power, or externalities. These market inefficiencies can lead to either underproduction or overproduction. Price ceilings and floors can create deadweight loss by causing excess supply or demand when the price is set above or below the market clearing price. Ignoring externalities, like pollution, can also lead to inefficient outcomes if social costs and benefits are not accounted for in supply and demand decisions. Monopolies restrict supply and raise prices above competitive levels, creating another source of deadweight loss.
This document summarizes key concepts from Chapter 4 of an International Economics textbook. It covers theories for trade protection such as the infant industry argument. It then discusses tariffs, including import/export tariffs and calculations of effective rates of protection. Finally, it examines nontariff barriers such as import quotas, tariff-rate quotas, and subsidies. For quotas and tariffs, it outlines the impacts on consumer surplus, producer surplus, and government revenue. The chapter suggests quotas impose larger losses than equivalent tariffs due to restrictions on consumption.
The theory of choice: Utility theory given uncertainty & State preference the...Md. Shohel Rana
Financial
Theory and
Corporate Policy
THOMAS E. COPELAND
Professor of Finance
University of California at Los Angeles
Firm Consultant, Finance
McKinsey & Company, Inc.
J. FRED WESTON
Cordner Professor of Managerial Economics and Finance
University of California at Los Angeles
This document discusses public goods and their efficient provision. It defines public goods as nonrival and nonexcludable, meaning consumption by one person does not reduce availability to others and it is difficult to prevent others from consuming them. This leads to a free rider problem where people have an incentive to let others pay for public goods while still enjoying the benefits. While private markets fail to efficiently provide public goods due to this issue, government intervention through taxation can potentially solve the free rider problem and lead to more efficient outcomes. The document also discusses the debate around privatizing certain goods and services traditionally provided publicly.
Non-Tariff barriers are trade barriers that restrict imports but are not in the usual form of a tariff.
Some common examples are anti-dumping measures and countervailing duties also called non-tariff barriers.
Non-Tariff barriers include macro-economic measures affecting trade.
Non-Tariff barriers comes under Trade Policy.
The document discusses Eurocurrency and the Eurodollar market. It defines Eurocurrency as currency deposited by governments or corporations in banks outside their home market, such as US dollars deposited in a London bank. The Eurodollar market refers specifically to US dollar deposits held in banks outside the US. The market originated in the late 1950s when European banks began accepting dollar deposits. It grew due to less regulation than in the US market, allowing for higher interest rates and more banking competition internationally. However, the unregulated nature of offshore banking also carries greater risks of bank failures and foreign exchange volatility for borrowers.
This document summarizes the key topics in international trade policy, including arguments for and against free trade. It discusses how free trade maximizes welfare but can have distributional effects. Arguments for free trade include efficiency gains and economies of scale. Arguments against include terms-of-trade gains from tariffs and addressing domestic market failures. Trade policy is also influenced by income distribution and political pressures. International trade agreements aim to liberalize trade through negotiation while balancing these various interests.
The document summarizes the evolution of international monetary systems between 1870-1973. It describes the gold standard period, the interwar years, the Bretton Woods system, and issues that arose. The Bretton Woods system established fixed exchange rates but collapsed in the early 1970s due to US inflation and balance of payments problems. The document analyzes policy options countries faced in pursuing internal and external balance under fixed exchange rates.
Aggregate demand is the total planned expenditure in an economy at a given price level. It is composed of consumption (C), investment (I), government spending (G), and net exports (X-M). Each component is influenced by various economic factors - consumption depends on disposable income and credit availability, investment depends on interest rates and profit levels, government spending depends on public opinion and the state of the economy, and net exports depend on domestic and foreign income and exchange rates. Changes in any component will cause the aggregate demand curve to shift right for an increase or left for a decrease on a graph with real output on the x-axis and price level on the y-axis.
What is interest ? Types of interest & Rate of interest
various theory of interest
1) Classical theory
2) Loanable fund theory
their criticism and explaination
The document discusses various trade policy instruments and their economic effects. It analyzes how tariffs, export subsidies, import quotas, and voluntary export restraints impact prices and trade volumes in importing and exporting countries. Tariffs reduce trade volume but generate government revenue, while quotas and export restraints reduce welfare by conferring quota rents on foreign producers. Local content rules pass higher costs onto domestic consumers.
This document discusses tariffs and their economic effects. It defines tariffs as taxes on imports and describes different types of tariffs such as ad valorem and specific tariffs. The document then analyzes the consumption, production, trade, and revenue effects of imposing a tariff using a partial equilibrium model. It also discusses the impact of tariffs on consumer and producer surplus. Finally, it provides an example comparing the effects of a tariff versus an import quota.
The document discusses regional economic integration and different levels of integration between countries. It defines various types of integration agreements like preferential trade areas, free trade areas, customs unions, common markets, and economic unions. It explains that preferential trade areas provide lower trade barriers between member countries than with non-members, while free trade areas remove all trade barriers but maintain external barriers. Customs unions remove internal barriers and adopt a common external trade policy. Common markets allow free movement of goods, services, labor and capital. Economic unions require harmonized economic and fiscal policies along with a common currency. Regional integration can bring economic and political benefits like increased trade, investment, market size and cooperation, but may also result in trade diversion and shifts in employment.
Open-Economy Macroeconomics: Basic ConceptsChris Thomas
This document provides an overview of key concepts in open-economy macroeconomics. It defines open and closed economies, and describes how an open economy interacts through international trade and financial flows. It explains exports, imports, the trade balance, and factors that influence them. It also discusses net capital flows, interest rates, and the relationship between saving, investment, and international flows. Finally, it introduces nominal and real exchange rates, and the theory of purchasing power parity.
This document discusses various aspects of inflation including definitions, types, measurement, causes and effects. It explains key inflation concepts like demand-pull and cost-push inflation. It also discusses the stages of inflation and inflation's relationship to GDP and currency valuation. The document provides examples of inflation rates in India and outlines some monetary and fiscal policy measures to control inflation.
The document discusses globalization and the international economy. It describes how globalization has occurred in waves due to decreases in trade barriers and transportation costs. More recently, developing countries have increased manufacturing trade and companies have outsourced production overseas. Several countries' levels of exports and imports as a percentage of GDP are provided as a measure of openness. The advantages of globalization are greater specialization and increased competition. However, some common misconceptions are that trade is zero-sum and import barriers create jobs.
A pure monopoly is characterized by a single seller of a unique product without close substitutes, the ability to set any price, and barriers to entry that prevent competition. While pure monopolies are rare, monopolies can exist when a firm is the only producer of a product and uses barriers like patents, large size, or unfair tactics to prevent competition. A monopoly has power over price as the sole provider of a product and sets price at the level that maximizes profits by balancing marginal revenue and marginal costs.
There are three main theories of exchange rate determination discussed in the document:
1) Purchasing power parity (PPP) theory states that exchange rates should adjust to equalize the prices of identical goods between countries when translated to a common currency.
2) Interest rate parity theory states that interest rate differentials between countries should equal expected changes in the exchange rate.
3) Balance of payments theory explains exchange rate movements through a country's balance of trade and capital flows.
Veblen goods, also known as conspicuous consumption goods, are luxury goods for which demand increases as the price increases, as it makes ownership a symbol of wealth and higher social status.
Exceptions to the law of demand include Veblen goods and Giffen goods. Veblen goods have a direct relationship between price and demand, while Giffen goods have an inverse relationship between income and demand for necessities.
Types of Giffen goods include necessary staple goods, inferior goods, and experience goods. For necessary staples and inferior goods, demand increases as prices rise due to consumers prioritizing basics. Experience goods see decreased demand if lower prices signal lower
Partial equilibrium, reference pricing and price distortion Devegowda S R
Partial equilibrium analysis examines how supply and demand interact in isolated markets. It assumes other markets are unchanged. A partial equilibrium exists when supply and demand are equal for a good, holding fixed other prices and incomes. This simplifies analysis but may not reflect real-world interactions between markets. Reference pricing refers to how consumers value a good relative to competitors' prices or a product's previous advertised price. Setting artificially high reference prices can mislead consumers into thinking they are getting a better deal. Price distortions occur when prices deviate from levels that would clear the market if all participants aimed to optimize returns. They can arise from inefficient transactions not aimed at return maximization.
This document discusses adversarial search in artificial intelligence. It provides an overview of games and introduces the minimax algorithm. The minimax algorithm is used to determine optimal strategies in two-player adversarial games by recursively considering all possible moves by both players. Tic-tac-toe is given as an example game where minimax can be applied to choose the best first move. The properties and limitations of the minimax algorithm are also summarized.
The theory of choice: Utility theory given uncertainty & State preference the...Md. Shohel Rana
Financial
Theory and
Corporate Policy
THOMAS E. COPELAND
Professor of Finance
University of California at Los Angeles
Firm Consultant, Finance
McKinsey & Company, Inc.
J. FRED WESTON
Cordner Professor of Managerial Economics and Finance
University of California at Los Angeles
This document discusses public goods and their efficient provision. It defines public goods as nonrival and nonexcludable, meaning consumption by one person does not reduce availability to others and it is difficult to prevent others from consuming them. This leads to a free rider problem where people have an incentive to let others pay for public goods while still enjoying the benefits. While private markets fail to efficiently provide public goods due to this issue, government intervention through taxation can potentially solve the free rider problem and lead to more efficient outcomes. The document also discusses the debate around privatizing certain goods and services traditionally provided publicly.
Non-Tariff barriers are trade barriers that restrict imports but are not in the usual form of a tariff.
Some common examples are anti-dumping measures and countervailing duties also called non-tariff barriers.
Non-Tariff barriers include macro-economic measures affecting trade.
Non-Tariff barriers comes under Trade Policy.
The document discusses Eurocurrency and the Eurodollar market. It defines Eurocurrency as currency deposited by governments or corporations in banks outside their home market, such as US dollars deposited in a London bank. The Eurodollar market refers specifically to US dollar deposits held in banks outside the US. The market originated in the late 1950s when European banks began accepting dollar deposits. It grew due to less regulation than in the US market, allowing for higher interest rates and more banking competition internationally. However, the unregulated nature of offshore banking also carries greater risks of bank failures and foreign exchange volatility for borrowers.
This document summarizes the key topics in international trade policy, including arguments for and against free trade. It discusses how free trade maximizes welfare but can have distributional effects. Arguments for free trade include efficiency gains and economies of scale. Arguments against include terms-of-trade gains from tariffs and addressing domestic market failures. Trade policy is also influenced by income distribution and political pressures. International trade agreements aim to liberalize trade through negotiation while balancing these various interests.
The document summarizes the evolution of international monetary systems between 1870-1973. It describes the gold standard period, the interwar years, the Bretton Woods system, and issues that arose. The Bretton Woods system established fixed exchange rates but collapsed in the early 1970s due to US inflation and balance of payments problems. The document analyzes policy options countries faced in pursuing internal and external balance under fixed exchange rates.
Aggregate demand is the total planned expenditure in an economy at a given price level. It is composed of consumption (C), investment (I), government spending (G), and net exports (X-M). Each component is influenced by various economic factors - consumption depends on disposable income and credit availability, investment depends on interest rates and profit levels, government spending depends on public opinion and the state of the economy, and net exports depend on domestic and foreign income and exchange rates. Changes in any component will cause the aggregate demand curve to shift right for an increase or left for a decrease on a graph with real output on the x-axis and price level on the y-axis.
What is interest ? Types of interest & Rate of interest
various theory of interest
1) Classical theory
2) Loanable fund theory
their criticism and explaination
The document discusses various trade policy instruments and their economic effects. It analyzes how tariffs, export subsidies, import quotas, and voluntary export restraints impact prices and trade volumes in importing and exporting countries. Tariffs reduce trade volume but generate government revenue, while quotas and export restraints reduce welfare by conferring quota rents on foreign producers. Local content rules pass higher costs onto domestic consumers.
This document discusses tariffs and their economic effects. It defines tariffs as taxes on imports and describes different types of tariffs such as ad valorem and specific tariffs. The document then analyzes the consumption, production, trade, and revenue effects of imposing a tariff using a partial equilibrium model. It also discusses the impact of tariffs on consumer and producer surplus. Finally, it provides an example comparing the effects of a tariff versus an import quota.
The document discusses regional economic integration and different levels of integration between countries. It defines various types of integration agreements like preferential trade areas, free trade areas, customs unions, common markets, and economic unions. It explains that preferential trade areas provide lower trade barriers between member countries than with non-members, while free trade areas remove all trade barriers but maintain external barriers. Customs unions remove internal barriers and adopt a common external trade policy. Common markets allow free movement of goods, services, labor and capital. Economic unions require harmonized economic and fiscal policies along with a common currency. Regional integration can bring economic and political benefits like increased trade, investment, market size and cooperation, but may also result in trade diversion and shifts in employment.
Open-Economy Macroeconomics: Basic ConceptsChris Thomas
This document provides an overview of key concepts in open-economy macroeconomics. It defines open and closed economies, and describes how an open economy interacts through international trade and financial flows. It explains exports, imports, the trade balance, and factors that influence them. It also discusses net capital flows, interest rates, and the relationship between saving, investment, and international flows. Finally, it introduces nominal and real exchange rates, and the theory of purchasing power parity.
This document discusses various aspects of inflation including definitions, types, measurement, causes and effects. It explains key inflation concepts like demand-pull and cost-push inflation. It also discusses the stages of inflation and inflation's relationship to GDP and currency valuation. The document provides examples of inflation rates in India and outlines some monetary and fiscal policy measures to control inflation.
The document discusses globalization and the international economy. It describes how globalization has occurred in waves due to decreases in trade barriers and transportation costs. More recently, developing countries have increased manufacturing trade and companies have outsourced production overseas. Several countries' levels of exports and imports as a percentage of GDP are provided as a measure of openness. The advantages of globalization are greater specialization and increased competition. However, some common misconceptions are that trade is zero-sum and import barriers create jobs.
A pure monopoly is characterized by a single seller of a unique product without close substitutes, the ability to set any price, and barriers to entry that prevent competition. While pure monopolies are rare, monopolies can exist when a firm is the only producer of a product and uses barriers like patents, large size, or unfair tactics to prevent competition. A monopoly has power over price as the sole provider of a product and sets price at the level that maximizes profits by balancing marginal revenue and marginal costs.
There are three main theories of exchange rate determination discussed in the document:
1) Purchasing power parity (PPP) theory states that exchange rates should adjust to equalize the prices of identical goods between countries when translated to a common currency.
2) Interest rate parity theory states that interest rate differentials between countries should equal expected changes in the exchange rate.
3) Balance of payments theory explains exchange rate movements through a country's balance of trade and capital flows.
Veblen goods, also known as conspicuous consumption goods, are luxury goods for which demand increases as the price increases, as it makes ownership a symbol of wealth and higher social status.
Exceptions to the law of demand include Veblen goods and Giffen goods. Veblen goods have a direct relationship between price and demand, while Giffen goods have an inverse relationship between income and demand for necessities.
Types of Giffen goods include necessary staple goods, inferior goods, and experience goods. For necessary staples and inferior goods, demand increases as prices rise due to consumers prioritizing basics. Experience goods see decreased demand if lower prices signal lower
Partial equilibrium, reference pricing and price distortion Devegowda S R
Partial equilibrium analysis examines how supply and demand interact in isolated markets. It assumes other markets are unchanged. A partial equilibrium exists when supply and demand are equal for a good, holding fixed other prices and incomes. This simplifies analysis but may not reflect real-world interactions between markets. Reference pricing refers to how consumers value a good relative to competitors' prices or a product's previous advertised price. Setting artificially high reference prices can mislead consumers into thinking they are getting a better deal. Price distortions occur when prices deviate from levels that would clear the market if all participants aimed to optimize returns. They can arise from inefficient transactions not aimed at return maximization.
This document discusses adversarial search in artificial intelligence. It provides an overview of games and introduces the minimax algorithm. The minimax algorithm is used to determine optimal strategies in two-player adversarial games by recursively considering all possible moves by both players. Tic-tac-toe is given as an example game where minimax can be applied to choose the best first move. The properties and limitations of the minimax algorithm are also summarized.
This document provides an introduction to using artificial intelligence for games. It discusses how AI can be used to create challenging opponents or helpful allies that act autonomously based on their programming. It notes that while human-level general intelligence is difficult to achieve, AI can perform well in narrow contexts like chess. For games, the AI must be intentionally flawed to ensure a fun challenge and cannot have obvious weaknesses. It must also be able to perform calculations and make decisions in real-time to interact with the game. The document then outlines some common AI techniques used in games, including MinMax search trees and finite state machines to control agent behavior. It provides pseudocode examples of the MinMax algorithm and discusses enhancements like Alpha-Beta pruning
Game Theory concepts are discussed including players, strategies, payoff tables, assumptions of rationality. Various game scenarios are presented involving dominance, saddle points, and mixed strategies. Graphical and linear programming solutions are provided for determining optimal strategies in games with mixed strategies where players' expected payoffs differ.
Game theory intro_and_questions_2009[1]evamstrauss
Here are the key steps:
Row
1. Check for dominant strategies - none exist
2. Find possible Nash equilibria by looking at best responses:
- Up, Left
- Down, Centre
3. Therefore, the two Nash equilibria are:
- (Up, Left)
- (Down, Centre)
The two Nash equilibria are (Up, Left) and (Down, Centre). There are no dominant strategies.
Is Enterprise Java Still Relevant (JavaOne 2015 session)Ian Robinson
Soon after Java burst into the world in the 90s it started to gatecrash the parties of its enterprise computing seniors, whose initial amused response was -- You're Not On The List, You're Not Coming In. But EJBs turned heads in the 20th Century and when the Java Enterprise platform emerged, it started getting more invites until it was the party. Now Java EE is grown up with its own kids - EE7 is already two years old. How is it and the platform doing? The party is now in the cloud and the guest list includes many different language technologies and fast-moving open-source innovations. Is Enterprise Java still relevant here? And if it is, what does it need to keep doing or what does it need to change to stay on the VIP list?
This document provides an introduction to using the TikZ package in LaTeX to create diagrams commonly used in economics. It includes examples of how to define coordinates, draw lines and arrows between coordinates, plot functions, shade areas, and more. The examples demonstrate how to draw a simple two-node network diagram and a normal distribution curve. TikZ allows for precise graphics generation directly in LaTeX documents.
Grzegorz Mazur is the co-boss and technical director at Vile Monarch. He discusses various techniques used for artificial intelligence in video games, including scripted AI, finite state machines, behavior trees, and influence mapping. Behavior trees allow for more complex and emergent agent behavior than finite state machines by evaluating decision trees. Influence mapping represents game elements as influence maps that can determine strategic areas for troop movements based on tensions between opposing forces.
The document discusses different classifications and concepts related to game theory, including:
1) Games can be zero-sum, where one player's gains equal another's losses, or non-zero-sum. They can involve 2 players or more.
2) Strategies can be pure, where players always choose the same action, or mixed, where players vary their actions randomly.
3) A payoff matrix outlines the potential payoffs for each combination of strategies between players in a 2-person, zero-sum game. The value of the game is when the maximin and minimax values are equal.
The document discusses various techniques used for artificial intelligence in gaming. It describes how state machines and planning systems are used to simulate human behavior for non-player characters. State machines define a character's states and transitions between states, but have limitations. Planning systems allow characters to work backwards from objectives to determine paths and behaviors. Additional techniques include navigation meshes to guide character movement and online learning from player data. The goal is to improve gaming experiences by making characters seem intelligent through these simulated human behavior methods.
The document discusses the Minimax algorithm and its application to game trees. It explains that Minimax is an optimal decision-making procedure for two-player zero-sum games where one player tries to maximize their score and the other tries to minimize it. It provides examples of how Minimax can be applied to games like Tic-Tac-Toe, Chess, Poker, and Monopoly to find the best move assuming the opponent plays optimally.
Reactive programming with Rx-Java allows building responsive systems that can handle varying workloads and failures. It promotes asynchronous and non-blocking code using observable sequences and operators. Rx-Java was created at Netflix to address issues like network chattiness and callback hell in their API. It transforms callback-based code into declarative pipelines. Key concepts are Observables that emit notifications, Operators that transform Observables, and Subscribers that receive emitted items. Rx-Java gained popularity due to its support for concurrency, error handling, and composability.
This document provides an overview of game theory and two-person zero-sum games. It defines key concepts such as players, strategies, payoffs, and classifications of games. It also describes the assumptions and solutions for pure strategy and mixed strategy games. Pure strategy games have a saddle point solution found using minimax and maximin rules. Mixed strategy games do not have a saddle point and require determining the optimal probabilities that players select each strategy.
Sun Certified Enterprise Architect Scea Mock ExamYasser Ibrahim
This document provides content from QuizOver.com related to the Oracle Certified Java Enterprise Architect certification. It includes sample questions covering topics like object-oriented design principles, architecture principles, integration protocols and messaging, security, and web and business tier technologies. The document provides the questions, potential answers, and links to the QuizOver website for the answer explanations. It also includes information about licensing the eBook content from QuizOver.com.
Game theory is the study of strategic decision making where outcomes depend on the choices of multiple players. It originated in the 1920s and was popularized by John von Neumann. Game theory analyzes cooperative and non-cooperative games with various properties like the number of players, information available, and whether choices are simultaneous or sequential. Important concepts in game theory include Nash equilibrium, where no player can benefit by changing strategy alone, and prisoner's dilemma, where defecting dominates but collective cooperation yields higher payoffs. Game theory is now used widely in economics, politics, biology, and other fields involving interdependent actors.
The document discusses combinations and permutations. It defines that a combination is when the order of items does not matter, while a permutation is when the order does matter. It provides examples of a fruit salad being a combination since the order of fruits does not change the salad, while a code combination is a permutation since the order is important. The document also shows how to calculate combinations and permutations using factorial notation, and provides examples of working through combination and permutation word problems.
The document provides information about getting help with programming homework assignments. It lists contact details for the programming homework help service, including a phone number, email address, and website URL. It then provides sample problems and solutions for dynamic programming algorithms. The problems cover topics like campaign strategy optimization, assigning animals to cages, counting paths in a graph, pizza partitioning, and stock price analysis. The solutions describe the subproblems, recursive relationships, base cases, and running times for dynamic programming algorithms to solve each problem in linear or quadratic time.
The document discusses Bayesian networks and how they can be used to concisely represent probability distributions over many variables by specifying conditional independence relationships between variables. It provides examples of how to construct Bayesian networks from probability distributions, how to perform inference by eliminating variables, and concepts like d-separation that characterize conditional independence in Bayesian networks.
This document discusses game theory and provides examples of different types of games. It introduces the prisoner's dilemma game, which involves two players who must choose whether to cooperate with or betray each other. It also discusses finding Nash equilibria, including examples of mixed strategy equilibria in the Battle of the Sexes and Matching Pennies games. The document provides information on concepts such as dominant strategies, Pareto optimality, best responses, and expected utility in game theory.
1. The document discusses using the binomial expansion and Stirling's formula to estimate the value of r that maximizes a binomial coefficient expression as n becomes large. Taking the limit as n approaches infinity, the optimal value of r is shown to be nq.
2. A example is given of estimating the probability of winning $40 or more by betting $1 on number 8 in roulette 500 times. Using the normal approximation, this probability is estimated to be about 25.8%.
This document contains solutions to 14 questions regarding permutations and combinations. It includes calculations of factorials, permutations of different numbers of items, and determining total possible arrangements under certain conditions. The questions involve finding the number of arrangements of letters, numbers, books, people and other items in different scenarios. Formulas for permutations and factorials are used throughout to calculate the total number of possible arrangements.
This document discusses Buss's theories of bounded arithmetic (S2i) and how they relate to the polynomial hierarchy (PH). It proposes using the separation of Buss's theories to approach the separation of levels of PH. It presents a consistency proof for S2i inside S2i+2 by introducing a predicate E for term existence and defining a bounded truth definition. The goal is to separate S2i and S2i+2 through Gödel's incompleteness theorem by showing S2i+2 can prove the i-consistency of S2i-E but S2i cannot prove its own consistency. Future work aims to simplify S2i-E and formally prove S2i can derive the i
Series solutions at ordinary point and regular singular pointvaibhav tailor
The document discusses series solutions for second order linear differential equations near ordinary and regular singular points.
It defines an ordinary point as a point where the functions p(x) and q(x) in the normalized form of the differential equation are analytic. Near an ordinary point, there exist two linearly independent power series solutions of the form Σcn(x-a)n that converge within the radii of convergence of p(x) and q(x).
It also discusses finding series solutions near a regular singular point x0=0, where the limits of p(x) and q(x) as x approaches 0 exist. An initial guess of a power series solution with exponent r is made, and the
This document discusses bounded arithmetic in free logic. It defines the theory S2iE, which is bounded arithmetic without truth values for terms. It proves that:
1) S2iE can "bootstrap" and prove basic properties, equality axioms, and predicate logic.
2) S2iE proves Σi-induction.
3) S2i-1E can be shown to be i-consistent within S2i+2E by defining a bounded truth definition and using a valuation tree and induction hypothesis.
This separation of S2i and S2i+1 theories using S2i-1E is a potential research direction.
- A function f from set A to B is one-to-one (1-1) if every element of B has at most one element of A that maps to it. The functions f(x)=x^2 from (-infinity,infinity) and g(x)=x^2 from [0,infinity) are examples, with g being 1-1 but f not.
- The inverse function f^-1 of a 1-1 function f maps each element of the range of f back to its unique preimage in the domain. For the function g above, g^-1(x)=sqrt(x).
- A set is countable if its elements can be mapped 1-1 and
Quadratic form and functional optimizationJunpei Tsuji
This slideshow describes a mathematics topic on quadratic form in English.
I made the slideshow to understand this topic on a deeper way.
I like linear algebra very much!!
This document summarizes methods for estimating average treatment effects in nonlinear models with endogenous switching using variably parametric regression. It describes two estimators: 1) a minimally parametric estimator that specifies an exponential conditional mean, and 2) a fully parametric estimator that specifies a generalized gamma conditional density. A Monte Carlo study shows the fully parametric estimator has lower bias even in small samples. The methods are then applied to a real dataset on birthweight and maternal smoking.
This document provides information about exponential functions:
- Exponential functions are defined as f(x) = ax, where a is a positive number called the base.
- The graphs of exponential functions with a base greater than 1 grow exponentially to the right, while graphs with a base between 0 and 1 slope down as they move to the right.
- Exponential functions are one-to-one and onto, appearing in processes like population growth, interest rates, and carbon dating.
Interpolation is a method for determining continuous values between discrete data points. Linear interpolation uses a weighted average to determine intermediate values between two known points along a straight line. Bilinear and bicubic interpolation generalize this concept to two and higher dimensions by interpolating between multiple known points. Interpolation has applications in image resizing, morphing, and synthesizing continuous values from discrete samples.
On Certain Classess of Multivalent Functions iosrjce
In this we defined certain analytic p-valent function with negative type denoted by 휏푝
. We obtained
sharp results concerning coefficient bounds, distortion theorem belonging to the class 휏푝
.
1) This document describes an optimal monetary policy model with 7 endogenous variables and 5 equilibrium conditions, leaving two degrees of freedom.
2) The model maximizes social welfare as the sum of period utilities from consumption and labor, subject to the equilibrium conditions.
3) The first order optimality conditions result in a system of 7 equations that can be solved using log-linearization methods around the non-stochastic steady state, similarly to previous examples.
1. This document provides an overview of key probability and statistics concepts covered on actuarial exams P and FM.
2. It covers topics like probability spaces, random variables, expectations, distributions, and functions including CDFs, PDFs, moments, and transformations.
3. Formulas and properties are presented for concepts like independence, conditional probability, multivariate distributions, the central limit theorem, and more.
The document provides an introduction and overview of the Python programming language including:
- Its origins and timeline from 1989 to present.
- How it combines functional, imperative and object-oriented paradigms.
- Details on dynamic vs static typing and how Python interprets source code.
- Benefits of its interactive shell, readability, large standard library and thriving community.
- Common uses like scripting, web development, science/engineering tasks, and jobs that utilize Python skills.
An introductory-to-mid level to presentation to complex network analysis: network metrics, analysis of online social networks, approximated algorithms, memorization issues, storage.
This document discusses social choice theory and group decision making. It covers topics like voting procedures, properties of voting systems, and impossibility theorems. Plurality voting is introduced as the simplest voting method where candidates are ranked first. Condorcet's paradox shows that no voting system can satisfy all voters. Arrow's theorem states that no voting system can satisfy certain desirable and reasonable criteria. Gibbard-Satterthwaite's theorem suggests that any voting system can be manipulated.
The document summarizes key concepts in social network analysis including metrics like degree distribution, path lengths, transitivity, and clustering coefficients. It also discusses models of network growth and structure like random graphs, small-world networks, and preferential attachment. Computational aspects of analyzing large networks like calculating shortest paths and the diameter are also covered.
This document discusses interoperability between Clojure and Python. It outlines Jython, which allows Python code to run on the Java Virtual Machine (JVM), and Clojure, a Lisp dialect that also runs on the JVM. It then covers how Clojure and Jython code can interact and call each other due to both running on the JVM. It provides examples of implementing Python objects in Clojure and calling Jython functions from Clojure.
The document discusses properties in Python classes. Properties allow accessing attributes through normal attribute syntax, while allowing custom behavior through getter and setter methods. This avoids directly accessing attributes and allows for validation in setters. Properties are defined using the @property and @setter decorators, providing a cleaner syntax than regular getter/setter methods. They behave like regular attributes but allow underlying method calls.
Simple presentation on Twisted fundamentals.
Originally part 4 of a 4 lectures seminar for the Networking class of the Computer Science course at the University of Parma
Object Oriented programming in Python.
Originally part 3 of a 4 lectures seminar for the Networking class of the Computer Science course at the University of Parma
A simple introduction to the Python programming language. In Italian. OLD: superseeded by Pycrashcourse 3.1.
Originally part 1 of a 4 lectures seminar for the Networking class of the Computer Science course at the University of Parma
A simple introduction to the Python programming language. In Italian. OLD: superseeded by Pycrashcourse 3.1.
Originally presented during the Networking class of the Computer Science course at the University of Parma
Object Oriented programming in Python.
Originally part 2 of a 4 lectures seminar for the Networking class of the Computer Science course at the University of Parma
Session 1 - Intro to Robotic Process Automation.pdfUiPathCommunity
👉 Check out our full 'Africa Series - Automation Student Developers (EN)' page to register for the full program:
https://bit.ly/Automation_Student_Kickstart
In this session, we shall introduce you to the world of automation, the UiPath Platform, and guide you on how to install and setup UiPath Studio on your Windows PC.
📕 Detailed agenda:
What is RPA? Benefits of RPA?
RPA Applications
The UiPath End-to-End Automation Platform
UiPath Studio CE Installation and Setup
💻 Extra training through UiPath Academy:
Introduction to Automation
UiPath Business Automation Platform
Explore automation development with UiPath Studio
👉 Register here for our upcoming Session 2 on June 20: Introduction to UiPath Studio Fundamentals: https://community.uipath.com/events/details/uipath-lagos-presents-session-2-introduction-to-uipath-studio-fundamentals/
[OReilly Superstream] Occupy the Space: A grassroots guide to engineering (an...Jason Yip
The typical problem in product engineering is not bad strategy, so much as “no strategy”. This leads to confusion, lack of motivation, and incoherent action. The next time you look for a strategy and find an empty space, instead of waiting for it to be filled, I will show you how to fill it in yourself. If you’re wrong, it forces a correction. If you’re right, it helps create focus. I’ll share how I’ve approached this in the past, both what works and lessons for what didn’t work so well.
In the realm of cybersecurity, offensive security practices act as a critical shield. By simulating real-world attacks in a controlled environment, these techniques expose vulnerabilities before malicious actors can exploit them. This proactive approach allows manufacturers to identify and fix weaknesses, significantly enhancing system security.
This presentation delves into the development of a system designed to mimic Galileo's Open Service signal using software-defined radio (SDR) technology. We'll begin with a foundational overview of both Global Navigation Satellite Systems (GNSS) and the intricacies of digital signal processing.
The presentation culminates in a live demonstration. We'll showcase the manipulation of Galileo's Open Service pilot signal, simulating an attack on various software and hardware systems. This practical demonstration serves to highlight the potential consequences of unaddressed vulnerabilities, emphasizing the importance of offensive security practices in safeguarding critical infrastructure.
From Natural Language to Structured Solr Queries using LLMsSease
This talk draws on experimentation to enable AI applications with Solr. One important use case is to use AI for better accessibility and discoverability of the data: while User eXperience techniques, lexical search improvements, and data harmonization can take organizations to a good level of accessibility, a structural (or “cognitive” gap) remains between the data user needs and the data producer constraints.
That is where AI – and most importantly, Natural Language Processing and Large Language Model techniques – could make a difference. This natural language, conversational engine could facilitate access and usage of the data leveraging the semantics of any data source.
The objective of the presentation is to propose a technical approach and a way forward to achieve this goal.
The key concept is to enable users to express their search queries in natural language, which the LLM then enriches, interprets, and translates into structured queries based on the Solr index’s metadata.
This approach leverages the LLM’s ability to understand the nuances of natural language and the structure of documents within Apache Solr.
The LLM acts as an intermediary agent, offering a transparent experience to users automatically and potentially uncovering relevant documents that conventional search methods might overlook. The presentation will include the results of this experimental work, lessons learned, best practices, and the scope of future work that should improve the approach and make it production-ready.
What is an RPA CoE? Session 1 – CoE VisionDianaGray10
In the first session, we will review the organization's vision and how this has an impact on the COE Structure.
Topics covered:
• The role of a steering committee
• How do the organization’s priorities determine CoE Structure?
Speaker:
Chris Bolin, Senior Intelligent Automation Architect Anika Systems
Must Know Postgres Extension for DBA and Developer during MigrationMydbops
Mydbops Opensource Database Meetup 16
Topic: Must-Know PostgreSQL Extensions for Developers and DBAs During Migration
Speaker: Deepak Mahto, Founder of DataCloudGaze Consulting
Date & Time: 8th June | 10 AM - 1 PM IST
Venue: Bangalore International Centre, Bangalore
Abstract: Discover how PostgreSQL extensions can be your secret weapon! This talk explores how key extensions enhance database capabilities and streamline the migration process for users moving from other relational databases like Oracle.
Key Takeaways:
* Learn about crucial extensions like oracle_fdw, pgtt, and pg_audit that ease migration complexities.
* Gain valuable strategies for implementing these extensions in PostgreSQL to achieve license freedom.
* Discover how these key extensions can empower both developers and DBAs during the migration process.
* Don't miss this chance to gain practical knowledge from an industry expert and stay updated on the latest open-source database trends.
Mydbops Managed Services specializes in taking the pain out of database management while optimizing performance. Since 2015, we have been providing top-notch support and assistance for the top three open-source databases: MySQL, MongoDB, and PostgreSQL.
Our team offers a wide range of services, including assistance, support, consulting, 24/7 operations, and expertise in all relevant technologies. We help organizations improve their database's performance, scalability, efficiency, and availability.
Contact us: info@mydbops.com
Visit: https://www.mydbops.com/
Follow us on LinkedIn: https://in.linkedin.com/company/mydbops
For more details and updates, please follow up the below links.
Meetup Page : https://www.meetup.com/mydbops-databa...
Twitter: https://twitter.com/mydbopsofficial
Blogs: https://www.mydbops.com/blog/
Facebook(Meta): https://www.facebook.com/mydbops/
"NATO Hackathon Winner: AI-Powered Drug Search", Taras KlobaFwdays
This is a session that details how PostgreSQL's features and Azure AI Services can be effectively used to significantly enhance the search functionality in any application.
In this session, we'll share insights on how we used PostgreSQL to facilitate precise searches across multiple fields in our mobile application. The techniques include using LIKE and ILIKE operators and integrating a trigram-based search to handle potential misspellings, thereby increasing the search accuracy.
We'll also discuss how the azure_ai extension on PostgreSQL databases in Azure and Azure AI Services were utilized to create vectors from user input, a feature beneficial when users wish to find specific items based on text prompts. While our application's case study involves a drug search, the techniques and principles shared in this session can be adapted to improve search functionality in a wide range of applications. Join us to learn how PostgreSQL and Azure AI can be harnessed to enhance your application's search capability.
LF Energy Webinar: Carbon Data Specifications: Mechanisms to Improve Data Acc...DanBrown980551
This LF Energy webinar took place June 20, 2024. It featured:
-Alex Thornton, LF Energy
-Hallie Cramer, Google
-Daniel Roesler, UtilityAPI
-Henry Richardson, WattTime
In response to the urgency and scale required to effectively address climate change, open source solutions offer significant potential for driving innovation and progress. Currently, there is a growing demand for standardization and interoperability in energy data and modeling. Open source standards and specifications within the energy sector can also alleviate challenges associated with data fragmentation, transparency, and accessibility. At the same time, it is crucial to consider privacy and security concerns throughout the development of open source platforms.
This webinar will delve into the motivations behind establishing LF Energy’s Carbon Data Specification Consortium. It will provide an overview of the draft specifications and the ongoing progress made by the respective working groups.
Three primary specifications will be discussed:
-Discovery and client registration, emphasizing transparent processes and secure and private access
-Customer data, centering around customer tariffs, bills, energy usage, and full consumption disclosure
-Power systems data, focusing on grid data, inclusive of transmission and distribution networks, generation, intergrid power flows, and market settlement data
Lee Barnes - Path to Becoming an Effective Test Automation Engineer.pdfleebarnesutopia
So… you want to become a Test Automation Engineer (or hire and develop one)? While there’s quite a bit of information available about important technical and tool skills to master, there’s not enough discussion around the path to becoming an effective Test Automation Engineer that knows how to add VALUE. In my experience this had led to a proliferation of engineers who are proficient with tools and building frameworks but have skill and knowledge gaps, especially in software testing, that reduce the value they deliver with test automation.
In this talk, Lee will share his lessons learned from over 30 years of working with, and mentoring, hundreds of Test Automation Engineers. Whether you’re looking to get started in test automation or just want to improve your trade, this talk will give you a solid foundation and roadmap for ensuring your test automation efforts continuously add value. This talk is equally valuable for both aspiring Test Automation Engineers and those managing them! All attendees will take away a set of key foundational knowledge and a high-level learning path for leveling up test automation skills and ensuring they add value to their organizations.
Conversational agents, or chatbots, are increasingly used to access all sorts of services using natural language. While open-domain chatbots - like ChatGPT - can converse on any topic, task-oriented chatbots - the focus of this paper - are designed for specific tasks, like booking a flight, obtaining customer support, or setting an appointment. Like any other software, task-oriented chatbots need to be properly tested, usually by defining and executing test scenarios (i.e., sequences of user-chatbot interactions). However, there is currently a lack of methods to quantify the completeness and strength of such test scenarios, which can lead to low-quality tests, and hence to buggy chatbots.
To fill this gap, we propose adapting mutation testing (MuT) for task-oriented chatbots. To this end, we introduce a set of mutation operators that emulate faults in chatbot designs, an architecture that enables MuT on chatbots built using heterogeneous technologies, and a practical realisation as an Eclipse plugin. Moreover, we evaluate the applicability, effectiveness and efficiency of our approach on open-source chatbots, with promising results.
AppSec PNW: Android and iOS Application Security with MobSFAjin Abraham
Mobile Security Framework - MobSF is a free and open source automated mobile application security testing environment designed to help security engineers, researchers, developers, and penetration testers to identify security vulnerabilities, malicious behaviours and privacy concerns in mobile applications using static and dynamic analysis. It supports all the popular mobile application binaries and source code formats built for Android and iOS devices. In addition to automated security assessment, it also offers an interactive testing environment to build and execute scenario based test/fuzz cases against the application.
This talk covers:
Using MobSF for static analysis of mobile applications.
Interactive dynamic security assessment of Android and iOS applications.
Solving Mobile app CTF challenges.
Reverse engineering and runtime analysis of Mobile malware.
How to shift left and integrate MobSF/mobsfscan SAST and DAST in your build pipeline.
How information systems are built or acquired puts information, which is what they should be about, in a secondary place. Our language adapted accordingly, and we no longer talk about information systems but applications. Applications evolved in a way to break data into diverse fragments, tightly coupled with applications and expensive to integrate. The result is technical debt, which is re-paid by taking even bigger "loans", resulting in an ever-increasing technical debt. Software engineering and procurement practices work in sync with market forces to maintain this trend. This talk demonstrates how natural this situation is. The question is: can something be done to reverse the trend?
"Frontline Battles with DDoS: Best practices and Lessons Learned", Igor IvaniukFwdays
At this talk we will discuss DDoS protection tools and best practices, discuss network architectures and what AWS has to offer. Also, we will look into one of the largest DDoS attacks on Ukrainian infrastructure that happened in February 2022. We'll see, what techniques helped to keep the web resources available for Ukrainians and how AWS improved DDoS protection for all customers based on Ukraine experience
The Department of Veteran Affairs (VA) invited Taylor Paschal, Knowledge & Information Management Consultant at Enterprise Knowledge, to speak at a Knowledge Management Lunch and Learn hosted on June 12, 2024. All Office of Administration staff were invited to attend and received professional development credit for participating in the voluntary event.
The objectives of the Lunch and Learn presentation were to:
- Review what KM ‘is’ and ‘isn’t’
- Understand the value of KM and the benefits of engaging
- Define and reflect on your “what’s in it for me?”
- Share actionable ways you can participate in Knowledge - - Capture & Transfer
inQuba Webinar Mastering Customer Journey Management with Dr Graham HillLizaNolte
HERE IS YOUR WEBINAR CONTENT! 'Mastering Customer Journey Management with Dr. Graham Hill'. We hope you find the webinar recording both insightful and enjoyable.
In this webinar, we explored essential aspects of Customer Journey Management and personalization. Here’s a summary of the key insights and topics discussed:
Key Takeaways:
Understanding the Customer Journey: Dr. Hill emphasized the importance of mapping and understanding the complete customer journey to identify touchpoints and opportunities for improvement.
Personalization Strategies: We discussed how to leverage data and insights to create personalized experiences that resonate with customers.
Technology Integration: Insights were shared on how inQuba’s advanced technology can streamline customer interactions and drive operational efficiency.
"What does it really mean for your system to be available, or how to define w...Fwdays
We will talk about system monitoring from a few different angles. We will start by covering the basics, then discuss SLOs, how to define them, and why understanding the business well is crucial for success in this exercise.
2. Introduction
Game theory can be defined as the study
of mathematical models of conflict and
cooperation between intelligent rational
decision-makers.
R. Myerson
3. Rationality and Intelligence
• A decision-maker is rational if he makes
decisions consistently in pursuit of his
own objectives
• A player is intelligent if he knows
everything that we know about the
game and he can make inferences about
the situation that we can make
4. Outcomes
• Let O be a finite set of outcomes
• A lottery is a probability distribution
over O l = [ p1 : o1 ,…, pk : ok ]
oi ∈O pi ∈[0,1]
k
∑p i =1
i=1
• We assume agents can rank outcomes
and lotteries with a utility function
5. TCP Game
Prisoner’s Dilemma
C D
C -1,-1 -4,0
D 0,-4 -3,-3
D: defective implementation
C: correct implementation
6. Game in Normal Form
• A finite n-person normal form game is a
tuple (N, A, u) where
• N is a finite set of n players
• A=A1⨉...⨉An, where Ai is a finite set of
actions available to player i
• u=(u1,...,un), where ui:A↦R is a real
valued utility function
• a=(a1,...,an) is an action profile
7. TCP Game (again)
A2 • N={1,2}
C D
• A={C,D}⨉{C,D}
C -1,-1 -4,0
D 0,-4 -3,-3 A1 A2 u1 u2
A1 C C -1 -1
C D -4 0
D C 0 -4
D D -3 -3
8. Actions
• Actions can be “arbitrarily
complex”
• Ex.: international draughts
• an action is not a move
• an action maps every
possible board configuration
to the move to be played if
the configuration occurs 2·1022
10. Rock-Paper-Scissor
Rock Paper Scissors
Rock 0,0 -1,1 1,-1
Paper 1,-1 0,0 -1,1
Scissors -1,1 1,-1 0,0
11. Strategy
• A pure strategy is selecting an action and playing it
• A mixed strategy for player i is an element of the set
Si=!(Ai) of probability distributions over Ai
• The support of a mixed strategy is the set of pure
strategies {ai|si(ai) > 0}
• The set of mixed-strategy profiles is S1⨉...⨉Sn and a
mixed strategy profile is a tuple (s1, ..., sn)
• The utility of a mixed strategy profile is
n
ui (s) = ∑ ui (a)∏ s j (a j )
a∈A j=1
12. Solution Concept
• Games are complex, the environment
can be stochastic, other player’s choices
affect the outcome
• Game theorists study certain subsets of
outcomes that are interesting in one
sense or another which are called
solution concepts
13. Pareto Efficiency
• The strategy profile s Pareto dominates
the strategy profile s’ if for some players
the utility for s is strictly higher and for
the others is not worse
• A strategy profile is Pareto optimal if
there is no other strategy profile
dominating it
14. Nash Equilibrium
• Player’s i best response to the strategy
profile s-i is a mixed strategy s*i ∈Si such
that ui(s*i ,s-i)≥ui(si, s-i) for every si ∈Si
• A strategy profile s=(s1,...,sn) is a Nash
equilibrium if, for all agents , si is a best
response to s-i
• Weak Nash (≥), Strong Nash (>)
15. Battle of the Sexes
LW WL • Both pure strategies are Nash
Equilibria
LW 2,1 0,0
• Are there any other Nash
WL 0,0 1,2 Equilibria?
• There is at least another
mixed-strategy equilibrium
(usually very tricky to
compute, but can be done with
simple examples)
16. Battle of the Sexes
LW WL • Suppose the husband chooses LW
with probability p and WL with
LW 2,1 0,0 probability p-1
• The wife should be indifferent
WL 0,0 1,2 between her available options,
otherwise she would be better off
choosing a pure strategy
Being “indifferent” means
• What is the p which allows the
obtaining the same utility, not wife to be really indifferent?
playing indifferently
• Please notice that the pure
strategies are Pareto optimal
17. Battle of the Sexes
p: probability that husband plays LW r: probability that wife plays LW
U wife (LW) = U wife (WL) U husband (LW) = U husband (WL)
1·p + 0·(1− p) = 0·p + 2·(1− p) 2·r + 0·(1− r) = 0·r + 1·(1− r)
2 1
p = 2− 2p p= 2r = 1− r r=
3 3
U w (s) = 2(1− p)(1− r) + pr
⎛ 2 1 ⎞ 2 2 2
U w ⎜ sw (r), LW + WL ⎟ = (1− r) + r =
⎝ 3 3 ⎠ 3 3 3
U h (s) = (1− p)(1− r) + 2 pr
⎛ 1 2 ⎞ 2 2 2
U h ⎜ sh ( p), LW + WL ⎟ = (1− p) + p =
⎝ 3 3 ⎠ 3 3 3
18. Matching Pennies
Heads Tails
Heads 1,-1 -1,1
• Do we have any pure
strategies?
Tails -1,1 1,-1 • No
• Do we have mixed
U1 (H) = U1 (T) strategies?
1·p + (−1)·(1− p) = −1·p + 1·(1− p)
• Yes
1
2 p − 1 = 1− 2 p p=
2
19. Rock-Paper-Scissor
Rock Paper Scissors
• Do we have any pure
Rock 0,0 -1,1 1,-1
strategies?
Paper 1,-1 0,0 -1,1
• No
Scissors -1,1 1,-1 0,0
• Do we have mixed
strategies?
pr + p p + ps = 1
• Yes
U1 (R) = U1 (P) = U1 (S)
20. Rock-Paper-Scissor
pr + p p + ps = 1 R P S
U1 (R) = U1 (P) = U1 (S) R 0,0 -1,1 1,-1
P 1,-1 0,0 -1,1
⎧0 pr + (−1) p p + 1ps = 1pr + 0 p p + (−1) ps S -1,1 1,-1 0,0
⎨
⎩1pr + 0 p p + (−1) ps = (−1) pr + 1p p + 0 ps
⎧ 1
ps + p p ⎪ ps =
⎧2 ps = pr + p p 2 ps = + pp 3
⎨ 2 ⎪
2 pr = ps + p p ⎪ 1
⎩ 4 ps = ps + 3p p ⎨ pr =
⎪ 3
ps = p p = pr ⎪p = 1
⎪ p 3
⎩
21. Existence of Nash Equilibria
• We have seen that not every game has a
pure strategy Nash equilibrium
• Does every game have a Nash
equilibrium (random or pure)?
Theorem (Nash, 1951) Every game with a
finite number of players and action profiles
has at least one Nash equilibrium
22. Computing Nash Equilibria
• There are algorithms which compute
Nash equilibria, but they are
exponential in the size of the game
• It is not known if there are polynomial
algorithms (but the consensus is that
there are none)
23. Dominated Strategies
Definitions
• Let si and si’ be two strategies of player i and S-i
the set of all strategy profiles of the remaining
players
• si strictly dominates si’ if for all s-i ∈S-i, it is the
case that ui(si,s-i)>ui(si’,s-i)
• si weakly dominates si’ if for all s-i ∈S-i, it is the
case that ui(si,s-i)≥ui(si’,s-i) and for at least one s-i
∈S-i it is the case that ui(si,s-i)>ui(si’,s-i)
• si very weakly dominates si’ if for all s-i ∈S-i, it
is the case that ui(si,s-i)≥ui(si’,s-i)
24. Dominated Strategies
Example
L C R L C
U 3,1 0,1 0,0 U 3,1 0,1
M 1,1 1,1 5,0 M 1,1 1,1
D 0,1 4,1 0,0 D 0,1 4,1
L C
U 3,1 0,1
D 0,1 4,1
25. Dominated Strategies
Prisoner’s Dilemma
C D C D
C -1,-1 -4,0 D 0,-4 -3,-3
D 0,-4 -3,-3
D
D -3,-3
26. Dominated Strategies
as solution concepts
• The set of all strategy profiles that assign
0 probability to playing any action that
would be removed through iterated
removal of strictly dominated strategies
is a solution concept
• Sometimes, no action can be “removed”,
sometimes we can solve the game (we
say the game is solvable by iterated
elimination)
27. Dominated Strategies
Costs
• Iterated elimination ends after a finite number of steps
• Iterated elimination preserves Nash equilibria
• We can use it to reduce the size of the game
• Iterated elimination of strictly dominated strategies can occur in
any order without changing the results
• Checking if a (possibly mixed) strategy is dominated can be
done in polynomial time
• Domination by pure strategies can be checked with a very
simple iterative algorithm
• Domination by mixed strategies can be checked solving a
linear problem
• Iterative elimination needs only to check pure strategies
28. Dominated Strategies
(domination by pure-strategies)
forall
pure
strategies
ai∈Ai
for
player
i
where
ai≠si
do
dom
←
true
forall
pure-‐strategy
profiles
a-‐i∈A-‐i
do
if
ui(si,a-‐i)≥ui(ai,a-‐i)
then
dom
←
false
break
if
dom
=
true
then
return
true
return
false
29. Other Solution Concepts:
Maxmin & Minmax
• The maxmin strategy for player i in an n player,
general sum game is a not necessarily unique (mixed)
strategy that maximizes i’s worst case payoff
• The maxmin value (or security level) is the minimum
payoff level guaranteed by a maxmin strategy
arg max si min s− i ui (si , s−i )
• In two player general sum games the minmax strategy
for player i against player –i is the strategy that keeps
the maximum payoff for –i at minimum
• It is a punishment
arg min si max s− i ui (si , s−i )
30. Other Solution Concepts:
Minmax, n-player
• In an n-player game, the minmax strategy for
player i against player j ≠ i is i’s component of the
mixed-strategy profile s-j in the expression:
arg min s− j max s j u j (s j , s− j )
where –j denotes the set of players other that j.
• Player i receives his minmax value if players –i
choose their strategies to minimize i utility “after”
he chose strategy si
• A player maxmin value is always less than or
equal to his minmax value
31. Maxmin & Minmax Examples
• Matching Pennies
• Maxmin: 0.5 T + 0.5 H
• Minmax: 0.5 T + 0.5 H
• Battle of Sexes
• Maxmin: H→0.66 LW + 0.33 WL
W→0.33 LW + 0.66 WL
• Minmax: H→0.66 LW + 0.33 WL
W→0.33 LW + 0.66 WL
32. Minmax Theorem
Theorem (von Neumann, 1928) In any finite,
two-player, zero-sum game, in any Nash
equilibrium each player receives a payoff that is
equal to both his maxmin value and his minmax
value
• Each player’s maxmin equals his minmax (value of the
game)
• Maxmin strategies coincide with minmax strategies
• Any maxmin strategy profile is a Nash equilibrium and
any Nash equilibrium is a maxmin strategy profile
34. Minimax Regret
• An agent i’s regret for playing an action ai if
other agents adopt action profile a-i is defined as:
⎡ ⎤
⎢ ai′ ∈Ai ui ( ai′ , a−i ) ⎥ − ui ( ai , a−i )
⎣
max
⎦
• An agent i’s maximum regret for playing an
action ai is defined as:
⎛⎡ ⎤ ⎞
max ⎜ ⎢ max ui ( ai′ , a−i ) ⎥ − ui ( ai , a−i )⎟
a− i ∈A− i ⎝ ⎣ ai ′ ∈Ai ⎦ ⎠
• Minimax regret actions for agent i are defined as:
⎡ ⎛⎡ ⎤ ⎞⎤
arg min ⎢ max ⎜ ⎢ max ui ( ai′ , a−i ) ⎥ − ui ( ai , a−i )⎟ ⎥
ai ∈Ai ⎣ a− i ∈A− i ⎝ ⎣ ai ′ ∈Ai ⎦ ⎠⎦
35. Maxmin vs. Minmax Regret
L R
regret(T ,[R])= 1− 1+ =
regret(B,[R]) = 1− 1 = 0
regret(T ,[L]) = 100 − 100 = 0 T 100, a 1-!, b
regret(B,[L]) = 100 − 2 = 98
max regret(T )= max{,0} = B 2, c 1, d
max regret(B) = max{98,0} = 98
P1 Maxmin is B (why?), his
Minimax Regret strategy is T
37. Computing Nash-Equilibrium
for two-players zero-sum games
• Consider the class of two-player zero-sum games
(
Γ = {1, 2} , A1 × A2 , ( u1 ,u2 ) )
*
• Ui is the expected utility for player i in equilibrium
• In the next slide, the LP for computing player 2 and
player 1 strategies are given
• Linear Programs are rather inexpensive to compute
38. Computing Nash-Equilibrium
for two-players zero-sum games
*
minimize U 1
subject to ∑ u (a , a ) ⋅ s
1 1
j k
2
k
2 ≤U
*
1 ∀j ∈A1
k∈A2
∑s k
2 =1
k∈A2
s ≥0
k
2 ∀k ∈A2
• Constants: u1(...)
• Variables: s2, U1*
U1* is a maxmin value!
39. Computing Nash-Equilibrium
for two-players zero-sum games
*
maximize U 1
subject to ∑ u (a , a ) ⋅ s
1 1
j k
2 1
j
≥U
*
1 ∀k ∈A2
j∈A1
∑s 1
j
=1
j∈A1
s ≥0
1
j
∀j ∈A1
• Constants: u1(...)
• Variables: s1, U1*
U1* is a maxmin value!
40. Computing Nash-Equilibrium
for two-players zero-sum games
*
minimize U 1
subject to ∑ u (a , a ) ⋅ s
1 1
j k
2
k
2 + r1 = U
j *
1 ∀j ∈A1
k∈A2
∑s k
2 =1
k∈A2
s ≥0
k
2 ∀k ∈A2
r1 ≥ 0
j
∀j ∈A1
• Constants: u1(...)
• Variables: s2, U1*
U1* is a maxmin value!
41. Computing Nash-Equilibrium
for two-players zero-sum games
*
minimize U 1
subject to ∑ u (a , a ) ⋅ s
1 1
j k
2 1
j
+r =U
2
k *
1 ∀k ∈A2
j∈A1
∑s 1
j
=1
j∈A1
s ≥0
1
j
∀j ∈A1
r ≥0
1
k
∀k ∈A2
• Constants: u1(...)
• Variables: s2, U1*
U1* is a maxmin value!
42. Computing maxmin & minmax
for two-players general-sum games
• We know how to compute minmax & maxmin strategies for
two-players 0-sum games
• It is sufficient to transform the general-sum game in a 0-sum
game
• Let G be an arbitrary two-player game G=({1,2}, A1⨉A2,(u1,u2));
we define G’= ({1,2}, A1⨉A2,(u1,-u1))
• G’ is 0-sum: a strategy that is part of a Nash equilibrium for
G’ is a maxmin strategy for 1 in G’
• Player 1 maxmin strategy is independent of u2
• Thus player’s 1 maxmin strategy is the same in G and in G’
• A minmax strategy for Player 2 in G’ is a minmax strategy
for 2 in G as well (for the same reasons)
43. Two Players General sum Games
• We can formulate the
∑ u (a , a ) ⋅ s
1 1
j k
2
k
2 + r1 = U
j *
1 ∀j ∈A1 game as a linear
complimentarity problem
k∈A2
∑ u (a , a ) ⋅ s
(LCP)
1 1
j k
2 1
j
+r =U2
k *
1 ∀k ∈A2
j∈A1
• This is a constraint
∑s 1
j
=1 ∑s k
2 =1 satisfaction problem
j∈A1 k∈A2 (feasibility, not
optimization)
s ≥ 0, s ≥ 0
1
j k
2 ∀j ∈A1 , ∀k ∈A2
r1 ≥ 0, r ≥ 0
j k
∀j ∈A1 , ∀k ∈A2 • The Lemke-Howson
2
algorithm is the best
r1 ·s = 0, r ·s = 0
j
1
j
2
k k
2 ∀j ∈A1 , ∀k ∈A2 suited to solve this kind
of problems
44. Computing n-players Nash
• Could be formulated as a nonlinear
complementarity problem (NLCP), thus it
would not be easily solvable
• A sequence of linear complementarity
problems (SLCP) can be used; it is not
always convergent, but if we’re lucky it’s fast
• It is possible to formulate as the
computation of the minimum of a specific
function, both with or without constraints
45. Software Tools
McKelvey, Richard D., McLennan,
Andrew M., and Turocy, Theodore L.
(2010). Gambit: Software Tools for Game
Theory, Version 0.2010.09.01.
http://www.gambit-project.org.
47. Extensive Form Game Example
Outline Introduction Game Representations Reductions
Extended Form of Game-1
• Each player bets a et
(2,-2)
me
coin 1.a raise 2.0
pa
ss
fo
ld
(1,-1)
• Player 1 draw a
(re 5
d)
.
(1,-1)
card and is the only 0
one to see it (-2,2)
et
.5 )
(b
me
la
ck
1.b raise 2.0
• Player 1 also plays pa
ss
before Player 2
fo
ld
(1,-1)
(-1,1)
48. Repeated Games
• What happens if the same NF game is
repeated:
• An infinite number of times?
• A finite number of times?
• A finite but unknown number of
times?
49. Bayesian Games
• Represent uncertainty about the game
being played; there is a set of possible
games
• with the same number of agents and
same strategy spaces but different
payoffs
• Agents beliefs are posteriors obtained
conditioning a common prior on
individual private signals