SYMBOLIC SYSTEMS 202: The Rationality Debate (3 units) Winter Quarter 2003-2004, Stanford University Instructor: Todd Davies

Game Theory Through Examples (2/11/04)

Games against nature - decision theory for a single agent

Expected utility theory for a single agent is sometimes called the theory of "games against nature". Consider this example.

Example 1: Planning a party

Our agent is planning a party, and is worried about whether it will rain or not. The utilities and probabilities for each state and action can be represented as follows: | | |Nature's states: | | | | |Rain |No rain | | | |(p=1/3) |(~p=2/3) | |Party planner's possible actions: |Outside |1 |3 | | |Inside |2 |2 |

The expected utility of an action A given uncertainty about a state S = Probability(S|A)*Utility(S|A) + Probability(not S|A)Utility(not S|A) Note that action A can be viewed as a compound gamble or outcome. Also, note that the probability of a state can depend on the agent's choice of action, although, in the above example, it does not.

For the party problem: EU(Outside) = (1/3)(1) + (2/3)(3) = 2.67; EU(Inside) = (1/3)(2) + (2/3)(2) = 2 Therefore, choose Outside, the action with the higher expected utility

(Noncooperative) game theory - decision theory for more than one agent, each acting autonomously (no binding agreements)

In the examples below, we'll assume two self-utility maximizing agents (or players), each of whom has complete information about the options available to themselves and the other player as well as their own and the other's payoffs (utilities) under each option.

Example 2 - Friends hoping to see each other

Consider two people, Chris and Kim. They both enjoy each other's company, but neither can communicate with the other before deciding whether to stay at home (where they would not see each other) or go to the beach this afternoon (where they could see each other). Each prefers going to the beach to being at home, and prefers being with the other person rather than being apart. This game can be represented by the following normal (or matrix) form:

Each player has a set of strategies (={Home,Beach} for both players in this example). Specifying one strategy i for the row player (Chris) and one strategy j for the column player (Kim) yields an outcome,...

...GameTheory
International Business Management
Preface
Since GameTheory is a tool used to analyze strategic behavior by taking into consideration how participants expect other to behave I thought about an everyday example in my life. I wanted to analyze my job at the bar and take into account some independent parties that compete with me. Since it’s not my choice who my boss will hire or fire I was interested in how each decisions outcome would be for the corresponding parties. What would be my benefits, considering different situations?
The game begins:
To finance my studies I work as a waitress in a bar. Usually, the bar is very crowded and it is difficult to serve all the guests. It is a tough job and sometimes very exhausting but serving all the guests is still doable. Nevertheless my boss realized that doable doesn’t mean high quality service. Due to the delay in receiving their order, customers sometimes complain to the boss. My boss is now in dilemma what to do. He could either replace me with a new waitress, hoping for better quality service with her, or he could hire a new waitress to support me in serving the customers and therefore have 2 waitresses serving. However, he knows the delay is most probably due to crowded bar and not due to lack of knowledge and skills form my side, he is having in mind to get a second waitress and have two waitresses,...

...NEGOTIATION
Negotiation theory
Last updated 9 months ago
The foundations of negotiation theory are decision analysis, behavioral decision making, gametheory, and negotiation analysis. Another classification of theories distinguishes between Structural Analysis, Strategic Analysis, Process Analysis, Integrative Analysis and behavioral analysis of negotiations.
Individuals should make separate, interactive decisions; and negotiation analysis considers how groups of reasonably bright individuals should and could make joint, collaborative decisions. These theories are interleaved and should be approached from the synthetic perspective.
Common Assumptions Of Most Theories
Negotiation is a specialized and formal version of conflict resolution most frequently employed when important issues must be agreed upon. Negotiation is necessary when one party requires the other party's agreement to achieve its aim. The aim of negotiating is to build a shared environment leading to long-term trust and often involves a third, neutral party to extract the issues from the emotions and keep the individuals concerned focused. It is a powerful method for resolving conflict and requires skill and experience. Zartman defines negotiation as "a process of combining conflicting positions into a common position under a decision rule of unanimity, a phenomenon in which the outcome is determined by...

...GAMESTHEORY
In gametheory, Nash equilibrium (named after John Forbes Nash, who proposed it) is a solution concept of a game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only his own strategy unilaterally. If each player has chosen a strategy and no player can benefit by changing his or her strategy while the other players keep theirs unchanged, then the current set of strategy choices and the corresponding payoffs constitute Nash equilibrium.
Stated simply, Amy and Phil are in Nash equilibrium if Amy is making the best decision she can, taking into account Phil's decision, and Phil is making the best decision he can, taking into account Amy's decision. Likewise, a group of players is in Nash equilibrium if each one is making the best decision that he or she can, taking into account the decisions of the others. However, Nash equilibrium does not necessarily mean the best payoff for all the players involved; in many cases, all the players might improve their payoffs if they could somehow agree on strategies different from the Nash equilibrium: e.g., competing businesses forming a cartel in order to increase their profits.
The prisoner's dilemma is a fundamental problem in gametheory that demonstrates why two people might not cooperate even if it...

...GameTheory in Economics
Gametheory is a concept of decision making that considers more elements beyond just benefits minus costs. Specifically, it includes the interaction between participants. In economics, the theory attempts to predict the participants’ optimal decisions. It has found a core place in economic decision-making and policy-making for its inherent ability to predict reactions in resource allocation, business negotiation, and other economic aspects. Gametheory is mostly associated with decision-theory and other contexts such as cooperation and negotiations.
From its definition, it is evident that the gametheory is largely used in the study of the human decision making processes. In psychology, its equivalent is known as the theory of social situations. In economics, however, gametheory tends to focus on sets of outcomes known as equilibrium that represent the most rational solutions to each situation. Gametheory emanates from the complexity of human interactions; thus, in a situation where an individual is dealing with an inanimate object such as a tree, he or she does not expect the tree to fight back or respond (Leyton-Brown and Shoham 51). The environment can also be considered neutral to what is done to the tree, at least in direct and...

...Beauty Contest Experiment
The experiment executed in the seminar was very simple. Players had to choose a number between 0 and 100. The objective is to choose a number based on your guess of the mean guesses of the group and multiply it by 2/3. It is called the Beauty contest Experiment because it was based on a theory John Maynard Keynes proposed on the relationship of the stock market with beauty contests conducted in newspapers of his time. In this report I will examine the logic behind choosing the best response strategy in theory and compare it with the actual results of the experiment conducted. From the comparison I will provide justification for why the theory is different from reality by also comparing it to examples in real life.
To understand the underlying logic of the game’s strategy one must understand the Nash Equilibrium. Princeton University’s Website (an excellent source since John Nash the person who came up with the Equilibrium attended that university) defines Nash Equilibrium as “a solution concept of a game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only his or her own strategy unilaterally. If each player has chosen a strategy and no player can benefit by changing his or her strategy while the other players keep theirs unchanged, then the current set of...

...implementation of directives has become significant concern in compliance discussion regarding the economic crisis in EU. According to Thomson (2009), directives refer to the important binding rules or enforceable legislative means proposed by the commission. The member countries of EU adopt the policies and legislations by compromised solutions after going through long and concentrated negotiations on the proposed legislations; the states have obligations to transpose the legislations and policies in their respective domestic laws.
The first section of the report entails the relevance of the compliance from the perspective of strategic interaction among the member states of EU. It will also describe the main actors along with the scenario. After that the solution concepts are selected in order to acquire a prediction about the possible behavior of the players and outcomes of their actions. These results will be obtained through implementation of the equilibrium concepts like gametheory selected for the current strategic framework and predictions will be made. Last section presents the discussion on the importance of results obtained through the implementation of the concept solutions.
Relevance from Strategic Perspective
As stated by the Thomson (2009), the most significant and enforceable resources are directives of the commission that are anticipated by the commission and the members states...

...SCHOOL OF BUSINESS AND ECONOMICS
ATHIRIVER CAMPUS
Table of Contents
GameTheory 3
History and impact of gametheory 5
Gametheory and information systems 6
Definition of key terms 6
Dominance 8
Nash equilibrium 8
Mixed strategies 9
Extensive games with perfect information 9
Extensive games with imperfect information 10
Zero-sum games and computation 11
Bidding in auctions 12
GameTheoryGametheory is the formal study of conflict and cooperation. Game theoretic concepts apply whenever the actions of several agents are interdependent. These agents may be individuals,
groups, firms, or any combination of these. The concepts of gametheory provide a language to formulate structure, analyze, and understand strategic scenarios.
The object of study in gametheory is the game, which is a formal model of an interactive situation. It typically involves several players; a game with only one player is usually called a decision problem. The formal definition lays out the players, their preferences, their information, the strategic actions available to them, and how these influence the outcome.
Games can be described formally at various levels of detail. A coalitional (or...

...Tutor: Xiang Sun∗ August 24, 2011 GameTheory
1
Review
• “Static” means one-shot, or simultaneous-move; “Complete information” means that the payoﬀ functions are common knowledge. • Normal-form representation: G = {S1 , . . . , Sn ; u1 , . . . , un }, where n is ﬁnite. • si is strictly dominated by si , if ui (si , s−i ) < ui (si , s−i ), ∀s−i ∈ S−i .
• Rational players do not play strictly dominated strategies, since they are always not optimal no matter what strategies others would choose. • Iterated elimination of strictly dominated strategies. This process is orderindependent. • Given other players’ strategies s−i ∈ S−i , Player i’s best response, denoted by Ri (s−i ), is the set of maximizers of maxsi ∈Si ui (si , s−i ), i.e., Ri (s−i ) = si ∈ Si : ui (si , s−i ) = max ui (si , s−i ) ⊂ Si .
si ∈Si
We call Ri the best-response correspondence for player i. • Given s−i , the best response Ri (s−i ) is a set. • In the n-player normal-form game G = {S1 , . . . , Sn ; u1 , . . . , un }, the strategy proﬁle (s∗ , . . . , s∗ ) is a pure-strategy Nash equilibrium if 1 n s∗ ∈ Ri (s∗ ), i −i equivalently, ui (s∗ , s∗ ) = max ui (si , s∗ ), −i i −i
si ∈Si
∀i = 1, . . . , n,
∀i = 1, . . . , n.
• {Nash equilibrium(a)} ⊂ {Outcomes of IESDS}.
∗
Email: xiangsun@nus.edu.sg; Mobile: 9169 7677; Oﬃce: S17-06-14.
1
MA4264 GameTheory
2/10
Solution to Tutorial 1
2
Tutorial...

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