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Questions tagged [game-theory]

Game theory is a study of situations of strategic interaction between two or more players in which there is a predefined set of rules and an outcome associated with each choice taken.

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Sequential Bertrand game with differentiated goods, how to write the strategies of firm 2

In a Bertrand competition with differentiated goods where firms set the prices sequentially, we have the following demand functions: q1 is quantity of goods demanded for firm 1 q2 is quantity of ...
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Rationalizable action profiles in nice symmetric games

Suppose we have a nice symmetric game with $n$ players, i.e. each player's action space is the same compact interval of the real line. I am tasked with identifying all of the rationalizable action ...
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Screening theory: Bundling problem

Consider an example of a risk-neutral seller that has two distinct indivisible goods for sale. The seller wants to maximize the expected revenue. The buyer's utility is $$I_av_a+I_bv_b-t,$$ where $I_a,...
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When solving game theory problems, how can you “assume” that the equilibrium strategy will lead to something happening?

This is something that confuses me a bit when I see my book solve game theory problems. When solving a model's equilibrium (like a subgame perfect nash equilibrium), the book will assume that a ...
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Is game theory supposed to be this easy? [on hold]

I have a course in game theory coming up, and I decided to look at the curriculum, which is a book by a guy called Gibbons on game theory. I skimmed it, but it seems that the core theory can be ...
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Game Theory Book

I am looking for a book on game theory, but I am not looking for a textbook. What I am after is more of a narrative form book that still provides valuable insights about game theory that I can pick up ...
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Can repeated game sustain a discounted average payoff that is not individually rational?

Look at the following payoff matrix, there are two stage Nash equlibria, and we consider a infinitely repeated game with discount factor $\delta\in(0,1)$, can players sustain an average payoff of (2,2)...
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What does these “strategy notations” mean?

In a sequential game, where there are 8 pizzas. Player 1 decides number of pizzas he wants. Let's call it S1 (strategy of player 1), and S1 = 5 means player one decided to get 5 pizzas. Then player ...
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Necessary indifference conditions in mixed equilibrium

Suppose we are playing a game where the Action set for Player 1 is $(a,b)$, for Player 2 is $(c,d)$, and for Player 3 is $(L,M,R)$. Assume that for Player 3, the action $M$ is weakly dominated by some ...
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Non-negative deviations from Nash Equilibrium

I know that in a Nash Equilibrium, no player can profitably deviate from the equilibrium strategy assuming that the strategies of the other players remain the same. My question is, what if a player ...
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286 views

Questions of the type “What do you think other people would think?”

For a study in the adoption of new technology, a student and I are developing a questionnaire that will poll domain experts on their opinions of what their colleagues would think about benefits/...
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Take It Or Leave It Strategy: Social Optimum

Here is what I understood Using Backward Induction, I inferred that buyer offers a price, say, $P$ and the seller will sell only if $P \geq c(I)$. Setting the lowest possible Price that will ensure ...
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Definition of Pareto efficiency and prisoner's dilemma

Pareto efficiency is defined in Wikipedia as: Pareto efficiency or Pareto optimality is a state of allocation of resources from which it is impossible to reallocate so as to make any one ...
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Finding correlated equilibria

I am a bit confused in my game theory class when it comes to finding correlated equilibria. I understand how to write the constraints using probability distributions, but I don't know how to find the ...
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How can I represent players with different information given continuous choice ranges?

I must represent in extended (tree) form the following Game Theory problem: A committee of experts must recommend a policy $a_1\in \mathbb R$ to Congress. Congress will then choose a policy $a_2\in ...
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Disagreement in Strategic Bargaining

Construct a pair of startegies for the ultimatum game ($T=1$ bargaining game), that constitutes a Nash Equilibrium and together support the outcome that there is no agreement reached by the two ...
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Mechanism Design: Proving that the expected utility is differentiable

Given a direct mechanism, we define a buyer's expected utility $u(\theta)$ conditional on her type being $\theta$ by $u(\theta)=\theta q(\theta)-t(\theta)$, where $q:[\underline{\theta},\bar{\theta}]\...
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Potential solutions of the $n$-player Prisoner's Dilemma

Suppose we want to model a real-world phenomenon as an $n$-player Prisoner's Dilemma. To be more specific, suppose we use RM Dawes' paper "The commons dilemma game: An n-person mixed-motive game with ...
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Global games: How to derive posterior with uniform prior and signal

I have access to some lecture notes on Global games (following the model of Carlsson and van Damme (1993)) showing how to derive the players posterior beliefs. But I don't really grasp how players ...
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Guess 2/3 of the average with integers - mixed strategy equilibria?

The "Guess 2/3 of the average" game (Wikipedia) is a game in which n people guess what 2/3 of the average of their guesses will be, and where the numbers are restricted to the integer (in this case) ...
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2answers
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Matchstick Game

Consider the following $2$ player finite extensive game. Player $1$ moves first. She must choose an integer from $1$ to $10$, and place that number of matchsticks on the table. Next, Player $2$ must ...
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Computing pure strategy Nash equilibria in finite games

I am trying to compute the (pure strategy) Nash equilibria of some discrete auctions. More precisely, let us define the strategy of each player as a function mapping from every valuation that they ...
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Strategy sets and SPNE

I need to find the startegy sets for the two players in this game. I understand that the game has only one subgame, that is, the entire game. I cannot seem to understand the strategy sets of the ...
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Pareto Efficiency Outcomes in Games

I have started studying the Pareto efficiency notion in Game theory. The definition I am familiar with is this: Strategy profile $\mathbf{s}$ Pareto dominates strategy $\mathbf{s}'$ if for all $i\in\...
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1answer
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Social Welfare and Pareto Optimality in a Bayesian Game

In a normal form game with two players, we call a joint strategy $s=(s_1,s_2)$ a Pareto optimal outcome if for no joint strategy $s'$, for all players $i\in\{1,2\}$, we have $$u_i(s')\geq u_i(s)\qquad(...
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1answer
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Understanding a multi-player Tragedy of the Commons game

In the formulation of the Tragedy of the Commons below, what does $n$ exactly represent? Is a threshold? The text is from plato.stanford.edu website on Kuhn's Prisoner's dilemma. Also, it says that $\...
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Mixed Strategies in Bayesian Games

I am confused about the following statement from Wikipedia: "A mixed strategy is an assignment of a probability to each pure strategy. This allows for a player to randomly select a pure strategy. ...
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Prisoner's dilemma as a Bayesian one-shot game

What happens if we assume that there is incomplete information to the prisoner's dilemma game? For example, suppose we have the following matrix with the utilities $T>R>P>S$ and $2R>S+T$ ...
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1answer
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Solving a two stage game by backward induction: which is the equilibrium notion?

Take a two-stage game with complete information and simultaneous actions in each state: (1) Player 1 and 2 simultaneously choose action $a_1\in A_1$ and $a_2\in A_2$ respectively. (2) Player 1 and ...
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1answer
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Static game with complete but imperfect information

I am confused on the concept of static game with complete but imperfect information and its consequences on the equilibrium definition. Suppose we have 2 players. Each player $i$ chooses action $Y_i\...
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If a mixed strategy is strictly dominated, then there is a strictly dominated pure strategy in its support?

I am looking at the proof of NE survives the iterated removal of strictly dominated strategies (MWG, ex 8.D.2) and in the solution manual, authors say something like if a mixed strategy is strictly ...
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Mixed Strategies — Does this lead to Nash equilibrium?

I have a simple payoff matrix defined here: https://prnt.sc/m6rp5m My question is: If both players play all 4 strategies with 1/4 probability, does that lead to nash equilibrium? I can't quite ...
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Nash Equilibria in Target Destroying-Guarding Game

Army A has a single plane which can strike one of three possible targets, A, B and C. Army B has one anti-aircraft gun that can be assigned to one of the three targets to guard it. The value of each ...
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Bayesian game and the set of types

In a finite Bayesian game, in most textbooks, a type of player $i$ is defined as $\theta_i\in\Theta_i$. Little is said about the "nature" of the set of types. For example, could we have a two-element ...
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Games with perfect but incomplete information in real life

It is hard to imagine such a game. Here is an example from wikipedia: Suppose you are playing a game of chess against an opponent who will be paid some substantial amount of money if a particular ...
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Best Response Functions for Cournot

For a Cournot Duopoly game with Linear Demand functions, we need to find the optimal quantity of each firm such that profit is maximised. Given the linearity of the Demand curves, we will obtain a ...
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1answer
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Definition of “simple situation” is ambiguous

"Simple” situations (where it is easy to predict which strategies will be chosen by the different agents): Each participant has a strategy that outperforms all others regardless of the ...
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Textbooks addressing MDP's in microeconomics/game theory/decision theory?

I have learned about the MDP formalism in the context of machine learning. I know that micro theorists also study MDP's, and agent behaviour in this context (both single and multi-agent I think). I ...
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1answer
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Rigorous solution of Stackelberg leader-follower game with N-players?

In the classical two-player Stackelberg game (sequential Cournot) we have a linear demand function $P = 1 - Q$ where $Q = \sum_0^2 q_i $, and we assume homogenous production cost $c$. By starting from ...
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1answer
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reference request: models of how racism or sexism affects decision making

My background is more in statistics and applied math, so I am not familiar with the breadth of decision making models from economics. However, I am working on a project that involves decision making ...
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2answers
103 views

Are direct mechanisms always truthful?

I am very new to mechanism design so I am a bit lost among the concepts, even the most basic ones. One thing that is not clear to me is the concept of direct mechanism and its relationship with the ...
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proof verification in George J. Mailath and Andrew Postlewaite (2001b)

I am reading George J. Mailath and Andrew Postlewaite (2001), and this is the lemma 1 in this paper. I don't understand the induction process in this proof. I think that we should use the fact that $...
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1answer
138 views

Perfect Bayesian Equilibria

What are the perfect Bayesian equilibria in the following game? The part that is confusing me is the right-hand side, where player 2 is of type N. Thanks! EDIT: My workings out so far are given ...
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1answer
37 views

Approximating common knowledge with common beliefs (Monderer and Samet, 1989)

I'm trying to understand Monderer and Samet's 1989 paper on approximate common knowledge. I'm stuck in the last part of the proof of the "agreeing to disagree" Theorem A, where the upper bound of the ...
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1answer
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What is the monotone hazard rate assumption used for?

I was reading this article on advance purchase discounts and pricing in which the author uses the monotone hazard rate assumption. Why does the author use it? The article is Advance-purchase ...
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Find a subgame perfect equilibrium and a Nash equilibrium

I want to know if my thinking is correct. Look at the following game. As the game has only one subgame (i.e., the game itself) then the Nash Equilibria will coincide with the subgame perfect ...
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Three stage two player signalling game, pooling equilbrium

I am solving a three-stage game in which first the sender sends the signal, then the receiver acts based on the signal and finally, the sender moves again. The sender has two types high and low and ...
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1answer
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How do I solve this Game Theory Bargaining problem? [closed]

Let's say there are two players in a game. In the first period, P1 makes an offer to split 3 dollars with P2. P2 accepts or rejects the offer. If P2 accepts, they get the offer made by P1, and P1 gets ...
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1answer
164 views

Cournot duopoly with differing costs

There are two firms in a Cournot duopoly that face inverse demand $P = \alpha - Q$, but one firm has total costs $c_1*q_1$ and the other has total costs $c_2*q_2$ with $c_1 < c_2$. I want to show ...
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1answer
402 views

All-Pay Auction Mixed Strategy Equilibrium

I am currently struggling with this exercise. Professor Nash announces that he will auction off a 20 dollars bill in a competition between two students chosen at random. Each student is to privately ...