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Bayesian Nash Equilibrium - UCLA Economics

Bayesian Nash EquilibriumIchiro ObaraUCLAF ebruary 1, 2012 Obara (UCLA) Bayesian Nash EquilibriumFebruary 1, 20121 / 28 Bayesian GameBayesian GameWe like to model situations where each party holds some privateinformation. For example,IA bidder does not know other bidders values in trader has some insider information about a recent technologicalinnovation by some (UCLA) Bayesian Nash EquilibriumFebruary 1, 20122 / 28 Bayesian GameBayesian GameBayesian GameABayesian Gameconsists ofplayer:a finite setNstate:a set action:a setAifor eachi Ntype:a setTifor eachi Nbelief:a functionpi:Ti ( j6=iTj)for eachi Npayoff:a functionui:A <for eachi (UCLA) Bayesian Nash EquilibriumFebruary 1, 20123 / 28 Bayesian GameBayesian GameInterpretation:I is a set of possiblestates of naturethat determine all physicalsetup of the game (payoffs).

Bayesian Nash equilibrium for the rst price auction It is a Bayesian Nash equilibrium for every bidder to follow the strategy b(v) = v R v 0 F(x)n 1dx F(v)n 1 for the rst price auction with i.i.d. private value. Obara (UCLA) Bayesian Nash Equilibrium February 1, 2012 17 / 28

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Transcription of Bayesian Nash Equilibrium - UCLA Economics

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Nash Equilibrium - University of California, Los Angeles

www.econ.ucla.edu

I Player 1’s best response against each of player 2’s strategy = red circle. I Player 2’s best response against each of player 1’s strategy = blue circle. I NE = (A,B),(B,A). Chicken Game 0, 0 4, 1 3, 3 AB 1, 4 A B Obara (UCLA) Nash Equilibrium January 11, 2012 9 / 31

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Dominant Strategy Equilibrium - UCLA Economics

www.econ.ucla.edu

Public Good Problem Consider the following situation. There is a plan to build a bridge in some village with n residents. Each resident’s bene t from the bridge is v i 2[0;v]. But this information is private. The cost of building a bridge is C > 0, which must be equally shared. Resident i’s payo is v i C=n if a bridge is built and 0 if not.

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Microeconomic Theory -1- Uncertainty - UCLA Econ

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Microeconomic Theory-3- Uncertainty © John Riley October 20, 2016 revised

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Repeated Game - UCLA Econ

www.econ.ucla.edu

Repeated game is a model about such long-term relationships. Obara (UCLA) Repeated Game March 1, 2012 2 / 33. Introduction A list of questions we are interested in: I When can people cooperate in a long-term relationship? I How do people cooperate? ... Repeated Game Ichiro Obara ...

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Note on Afriat™s Theorem, Ichiro Obara, September 29, 2009

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Note on Afriat™s Theorem, Ichiro Obara, September 29, 2009 This note is based on Fostel, Scarf, and Todd (2004). Consider a –nite data set D =

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John Riley minor corrections 25 July 2016

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2. Concave function of one variable 4 3. Concave function of more than one variable 7 4. Necessary and sufficient conditions for a maximum 10 5. When is a function concave 11 6. The gains to diversifying 15 7. Production plans and supporting …

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Cost Functions - UCLA Economics

www.econ.ucla.edu

• Suppose firm wishes to raise production – Can’t change capital stock – Hires more workers. – Capital/Labor balance no longer optimal. – High production costs. 35 Time Frames • In very short run, ... startup cost of (r 1 +r 2) is sunk. • Cost function is thus with c(r 1,r 2,0) = r 1 +r 2.

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Preference and Utility - UCLA Economics

www.econ.ucla.edu

formulate the consumer problem as a constrained optimization problem: max x2X u(x) s:t:p x w; or equivalently, max x2B(p;w) u(x), which may be easily solved analytically or numerically. Obara (UCLA) Preference and Utility October 2, 2012 12 / 20

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Pure Strategy Nash Equilibrium A strategy vector s = (s 1;:::;s k) is a pure strategy Nash Equilibrium (pure Nash) if c i (s) c i(s0;s i) for all i, and for all s0 i 2S i. Intuitively, no player is able to decrease their cost through unilateral action (choosing another of their strategies while everybody else remains the same).

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Nash Equilibrium - University of California, Los Angeles

www.econ.ucla.edu

Nash equilibrium for a strategic game is a pro le of actions such that each action is a best response to the other actions. Let B i(a i) ˆA i be the set of player i’s best response actions against a i 2A i. Here is the formal de nition of Nash equilibrium. Nash Equilibrium a = (a 1

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Nash

library.princeton.edu

Title: Nash.pdf Created Date: 12/11/2001 4:05:15 PM

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