3. Basic Topics in Game Theory

Strategic Behavior in Business and Econ Outline 3.1 What is a Game ? The elements of a Game The Rules of the Game: Example Examples of Game Situations Types of Games : Complete and Incomplete Infomation 3.2 Solution Concepts Static Games of complete information: Dominant Strategies and Nash Equilibrium in pure and mixed strategies Dynamic Games of complete information: Backward Induction and Subgame perfection

Strategic Behavior in Business and Econ 3.1. What is a Game ?

Strategic Behavior in Business and Econ Outline 3.1 What is a Game ? The elements of a Game The Rules of the Game: Example Examples of Game Situations Types of Games : Complete and Incomplete Infomation 3.2 Solution Concepts Static Games of complete information: Dominant Strategies and Nash Equilibrium in pure and mixed strategies Dynamic Games of complete information: Backward Induction and Subgame perfection

Strategic Behavior in Business and Econ So far we have studied Decision Problems. Those were scenarios where the final outcome of our decision depends on both the decision we take and (maybe) some uncertain events. Now we are going to study Strategic Problems. These are scenarios where the final outcome of our decision depends on both the decision we take and also the decision taken by other agents (and maybe some uncertain events) Such scenarios are often called “Games”

Strategic Behavior in Business and Econ Game and Decision Th. Social Choice Theory Game Theory Decision Theory Decision Th. Under Cert. Decision Th. Under Uncert. Decision Problem Individual Decision Collective Decision Unipersonal Without Risk With Risk Multipersonal

Strategic Behavior in Business and Econ What is a Game ? A Game is an scenario in which several agents (the players) take decisions (follow strategies) that affect each other's final outcome (the payoffs) What is Game Theory ? Game Theory is a set of techniques that help in the analysis and understanding of game situations and, in many cases, can produce a prediction of what is the more “logical” outcome of the game

Strategic Behavior in Business and Econ “Game theory doesn’t give you answers. But it does help you discipline your intuition, requiring you to first be explicit about your assumptions and then helping you see their interaction without allowing extraneous considerations to cloud the picture. So think of game theory (or any formal modeling) as Photoshop: it can remove the red1eye and clear up the resolution, but you still take the picture.” Nathan Griffith PhD. Political Science Belmont University

Strategic Behavior in Business and Econ Game Theory does not teach you how to win ! (But your chances of winning may improve if you use Game Theory)

Strategic Behavior in Business and Econ The Elements of a Game

Strategic Behavior in Business and Econ Outline 3.1 What is a Game ? The elements of a Game The Rules of the Game: Example Examples of Game Situations Types of Games : Complete and Incomplete Infomation 3.2 Solution Concepts Static Games of complete information: Dominant Strategies and Nash Equilibrium in pure and mixed strategies Dynamic Games of complete information: Backward Induction and Subgame perfection

Strategic Behavior in Business and Econ The elements of a Game 1.The environment of the game Players Strategies Payoffs 2.The rules of the game Timing of moves Nature of conflict and interaction Information conditions 3.The assumptions of game theory Rationality Common knowledge For each different game we must identify its environment (1) and understand the rules (2). The assumptions (3) are “simplifications” that apply to any game

Strategic Behavior in Business and Econ Players Everyone who has to take an action that has an effect on your earnings All players' earnings are also affected by your decision Strategies Actions available to each player Define a plan of action for every contingency Payoffs “Numbers” associated with each outcome Reflect the “utility” of the players 1. The environment of the game

Strategic Behavior in Business and Econ 2. The rules of the game Timing of moves Are decisions taken simultaneously or sequentially (one after the other) ? Nature of conflict and interaction Are players’ interests in conflict? Do they seek to cooperate or coordinate ? Will players interact once or repeatedly? Information conditions Do all players have the same information ? Do players always know the “history” of the game ?

Strategic Behavior in Business and Econ 3. The assumptions of game theory Rationality Players aim to maximize their payoffs They are selfish, they only care about their payoffs Players are perfect calculators Common Knowledge Each player knows the environment and the rules of the game Each player knows that each player knows the environment and the rules Each player knows that each player knows that each player knows the environment and the rules Each player knows that each player knows that each player knows that each player knows the environment and the rules Etc. Etc. Etc.

Strategic Behavior in Business and Econ 1.The environment of the game Players Strategies Payoffs For any specific game, these have to be clearly identified. 1.The rules of the game Timing of moves Nature of conflict and interaction Information conditions The rules of the game need to be understood, but they can not be changed 1.The assumptions of game theory Rationality Common knowledge

Strategic Behavior in Business and Econ Example: The Obama-McCain debate at the University of Mississippi (by N. Griffith) So let’s look at McCain’s situation. There are two players, McCain and Obama. They have two choices: attend the Mississippi debate or not. So there are four potential outcomes: 1. Both attend the debate. 2. Obama attends the debate, but McCain does not. 3. McCain attends the debate, but Obama does not. 4. Both skip the debate to attend to other (allegedly more pressing) matters. McCain preferences: 4 better than 1 better than 2 better than 3 Obama preferences: 2 better than 1 better than 4 better than 3

Strategic Behavior in Business and Econ Example: The Obama-McCain debate at the University of Mississippi (by N. Griffith) We need to identify: The players:Obama and McCain The strategies:Attend and Do not attend (both players have the same strategies) The payoffs:For each player, we can define a payoff of 4 to the best outcome, 3 to the second best, 2 to the third and 1 to the worst outcome of all (based on the preferences of each one) [ see table on next slide ] and to understand the rules of the game Timing of moves:They have to decide without knowing the decision of the other (simultaneous move) Nature of conflict and interaction:They have opposing interests (conflict) Information conditions: The two of them have the same information on all the items above

Strategic Behavior in Business and Econ Attend Do not attend Mr. McCain (player 1) Mr. Obama (player 2) 3, 3 Example: The Obama-McCain debate at the University of Mississippi (by N. Griffith) Attend Do not attend 1, 1 4, 22, 4

Strategic Behavior in Business and Econ Attend Do not attend Mr. McCain (player 1) Mr. Obama (player 2) 3, 3 Example: The Obama-McCain debate at the University of Mississippi (by N. Griffith) Attend Do not attend 1, 1 4, 22, 4 Prediction of Game Theory: Obama's best strategy is clear Attend no matter what

Strategic Behavior in Business and Econ Attend Do not attend Mr. McCain (player 1) Mr. Obama (player 2) 3, 3 Example: The Obama-McCain debate at the University of Mississippi (by N. Griffith) Attend Do not attend 1, 1 4, 22, 4 Prediction of Game Theory: Obama's best strategy is clear Attend no matter what

Strategic Behavior in Business and Econ Attend Do not attend Mr. McCain (player 1) Mr. Obama (player 2) 3, 3 Example: The Obama-McCain debate at the University of Mississippi (by N. Griffith) Attend Do not attend 1, 1 4, 22, 4 Prediction of Game Theory: McCain's best strategy is not so clear Attend if Obama does Do not Attend otherwise

Strategic Behavior in Business and Econ Attend Do not attend Mr. McCain (player 1) Mr. Obama (player 2) 3, 3 Example: The Obama-McCain debate at the University of Mississippi (by N. Griffith) Attend Do not attend 1, 1 4, 22, 4 Prediction of Game Theory: McCain's best strategy is not so clear Attend if Obama does Do not Attend otherwise

Strategic Behavior in Business and Econ Attend Do not attend Mr. McCain (player 1) Mr. Obama (player 2) 3, 3 Example: The Obama-McCain debate at the University of Mississippi (by N. Griffith) Attend Do not attend 1, 1 4, 22, 4 Prediction of Game Theory: But McCain's is rational, and he knows Obama is also rational. Hence he knows that Obama will Attend

Strategic Behavior in Business and Econ Attend Do not attend Mr. McCain (player 1) Mr. Obama (player 2) 3, 3 Example: The Obama-McCain debate at the University of Mississippi (by N. Griffith) Attend Do not attend 1, 1 4, 22, 4 Prediction of Game Theory: But McCain's is rational, and he knows Obama is also rational. Hence he knows that Obama will Attend

Strategic Behavior in Business and Econ Attend Do not attend Mr. McCain (player 1) Mr. Obama (player 2) 3, 3 Example: The Obama-McCain debate at the University of Mississippi (by N. Griffith) Attend Do not attend 1, 1 4, 22, 4 Prediction of Game Theory: Therefore, McCain's best strategy is also to Attend

Strategic Behavior in Business and Econ Attend Do not attend Mr. McCain (player 1) Mr. Obama (player 2) 3, 3 Example: The Obama-McCain debate at the University of Mississippi (by N. Griffith) Attend Do not attend 1, 1 4, 22, 4 Prediction of Game Theory: Therefore, McCain's best strategy is also to Attend

Strategic Behavior in Business and Econ Example: The Obama-McCain debate at the University of Mississippi (by N. Griffith) On the payoffs: In this example the payoffs have been chosen arbitrarily 4 to the best outcome 3 to the second best etc The specific value of these payoffs is not important as long as they keep the same order as in the actual preferences, that is, Highest payoff to the best outcome Second highest payoff to the second best outcome etc

Strategic Behavior in Business and Econ Example: The Obama-McCain debate at the University of Mississippi (by N. Griffith) You may check that, in this example, the final prediction of Game Theory remains the same if we change the payoff values as in the table below In many games the payoffs are given, and correspond to real values (monetary profits, market share, expected value of a lottery, etc). In other cases, as in this example, we must define the payoffs so that they are consistent with the preferences of the players (ordinal payoffs)

Strategic Behavior in Business and Econ The Rules of the Game

Strategic Behavior in Business and Econ Outline 3.1 What is a Game ? The elements of a Game The Rules of the Game: Example Examples of Game Situations Types of Games : Complete and Incomplete Infomation 3.2 Solution Concepts Static Games of complete information: Dominant Strategies and Nash Equilibrium in pure and mixed strategies Dynamic Games of complete information: Backward Induction and Subgame perfection

Strategic Behavior in Business and Econ As said before, the rules of the game are usually fixed and they can not be changed. We must be aware of them in order to understand how the game works, and to be able to perform a proper analysis. Nevertheless, there are cases in which some manipulation of the rules is possible. Then, it might be profitable to shape the rules to our own benefit

Strategic Behavior in Business and Econ Example: The Agenda Setting The city government of a small town must decide (by majority voting) the location of a new school. There are 3 possible locations available: 1. Next to the old school (OS) 2. Next to the Chapel (CH) 3. Next to the city park (CP) Because of their background, the preferences of the nine member of the city council are known to everybody Four prefer OS to CH to CP Three prefer CH to CP to OS Two preferCP to OS to CH

Strategic Behavior in Business and Econ Example: The Agenda Setting By Majority Voting (an alternative wins if it obtains more than 50% of the votes), there is no winner OS gets 4 votes CH gets 3 votes(At least 5 votes are CP gets 2 votes needed to win) So, they decide to vote first between two alternatives and then vote between the winner and the third alternative. The Mayor must set the voting agenda (or voting rule)

Strategic Behavior in Business and Econ Example: The Agenda Setting Agenda 1: First, OS vs. CH Then, winner vs CP Agenda 2: First, OS vs. CP Then, winner vs CH Agenda 3: First, CP vs. CH Then, winner vs OS Imagine you are the Mayor, what Agenda would you choose ?

Strategic Behavior in Business and Econ Example: The Agenda Setting Agenda 1: First, OS vs. CH OS (gets 6 votes) Then, OS vs CP CP (gets 5 votes) Agenda 2: First, OS vs. CP CP (gets 5 votes) Then, CP vs CH CH (gets 7 votes) Agenda 3: First, CP vs. CH CH (gets 7 votes) Then, CH vs OS OS (gets 6 votes) What will be the winner in each case ?

Strategic Behavior in Business and Econ Example: The Agenda Setting Thus, if you are the Mayor, depending on whether you prefer OS, CH, or CP, you will set the corresponding Agenda for voting In this sense, some rules are “manipulable”

Strategic Behavior in Business and Econ Example: The Agenda Setting But you could also “manipulate” the outcome of the game without being the Mayor ! That is, you can change the outcome of the game without changing nor manipulating the rules of the game ! How ?

Strategic Behavior in Business and Econ Example: The Agenda Setting For instance, imagine that the Mayor sets Agenda 2 (because she likes CH) Agenda 2: First, OS vs. CP CP (gets 5 votes) Then, CP vs CH CH (gets 7 votes) and you are one of the two people that have CH as the worst alternative What can you do ?

Strategic Behavior in Business and Econ Example: The Agenda Setting Lie !

Strategic Behavior in Business and Econ Example: The Agenda Setting When you are called to cast your vote in the first round (OS vs. CP), you should vote for OS (even though CP is your preferred alternative). Then OS will win the first round ! (will get 5 votes) Then, in the second round, when voting between OS and CH, you can vote truthfully and OS will win the final round with 6 votes. Agenda 2: First, OS vs. CP OS (gets 5 votes) Then, OS vs CH OS (gets 7 votes) So, you changed the outcome of the game without changing the rules !

Strategic Behavior in Business and Econ Summary A Game is a involved scenario where the profit of each player depends on the actions of all the players Make sure to clearly identify the players, the strategies, and the payoffs Make sure to understand the rules of the game Make sure the rules are not manipulable Act rational and always think that other players are rational as well Never take the behavior of the other players as given