network effects - columbia universityac3827/codesslides/networkeffects.pdfnetwork effects captured...
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![Page 1: Network Effects - Columbia Universityac3827/CodesSlides/NetworkEffects.pdfNetwork effects captured one widely observed phenomenon of the behavior of agents in a network, especially](https://reader030.vdocument.in/reader030/viewer/2022041014/5ec549b01146c40e7c44960e/html5/thumbnails/1.jpg)
Network Effects
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Motivation01
CONTENTS Network Effect Recap02
Program Demo03
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MOTIVATION
01PART
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MOTIVATION
Ideal Model for ImplementationThe majority of the analysis is based on the generic form of the utility function: π’ π π,π‘ , π₯π,π‘ , ππ‘ , ππ‘β1 =
π₯π πβ ππ‘β1 + 1 β π β ππ‘ π π,π‘ β ππ π,π‘ and the distribution πΊ(π₯)
Key Characteristic of NetworksNetwork effects captured one widely observed phenomenon of the behavior of agents in a network,
especially in social network settings
Intuitive Study ToolA graphical representation of the result we got in class will be a very intuitive study tool for
students to better understand the theories on network effect
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NETWORK EFFECTRECAP
02PART
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NETWORK EFFECT RECAP β SET UP
Infinite AgentsModelling a community with a large number of agents. For instance, a continuum πΌ β [0,1] of agents
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NETWORK EFFECT RECAP β SET UP
Homogeneous Utility FunctionEach agent has the utility function: π’ π π,π‘ , π₯π,, ππ‘ , ππ‘β1 = π₯π πβ ππ‘β1 + 1 β π β ππ‘ β π π π,π‘, where π π,π‘ β {0,1}
denotes the strategy of agent π at time π‘, and ππ‘ denotes the fraction of population that choose π = 1
Infinite AgentsModelling a community with a large number of agents. For instance, a continuum πΌ β [0,1] of agents
![Page 8: Network Effects - Columbia Universityac3827/CodesSlides/NetworkEffects.pdfNetwork effects captured one widely observed phenomenon of the behavior of agents in a network, especially](https://reader030.vdocument.in/reader030/viewer/2022041014/5ec549b01146c40e7c44960e/html5/thumbnails/8.jpg)
NETWORK EFFECT RECAP β SET UP
Homogeneous Utility FunctionEach agent has the utility function: π’ π π,π‘ , π₯π , ππ‘ , ππ‘β1 = π₯π πβ ππ‘β1 + 1 β π β ππ‘ β π π π,π‘, where π π,π‘ β {0,1}
denotes the strategy of agent π at time π‘, and ππ‘ denotes the fraction of population that choose π = 1
Type DistributionEach agent has a type π₯π, which describes his/her utility from taking action π = 1. π₯π comes from the
distribution πΊ(π₯) (CDF)
Infinite AgentsModelling a community with a large number of agents. For instance, a continuum πΌ β [0,1] of agents
![Page 9: Network Effects - Columbia Universityac3827/CodesSlides/NetworkEffects.pdfNetwork effects captured one widely observed phenomenon of the behavior of agents in a network, especially](https://reader030.vdocument.in/reader030/viewer/2022041014/5ec549b01146c40e7c44960e/html5/thumbnails/9.jpg)
NETWORK EFFECT RECAP β SET UP
Homogeneous Utility FunctionEach agent has the utility function: π’ π π,π‘ , π₯π , ππ‘ , ππ‘β1 = π₯π πβ ππ‘β1 + 1 β π β ππ‘ β π π π,π‘, where π π,π‘ β {0,1}
denotes the strategy of agent π at time π‘, and ππ‘ denotes the fraction of population that choose π = 1
Type DistributionEach agent has a type π₯π, which describes his/her utility from taking action π = 1. π₯π comes from the
distribution πΊ(π₯) (CDF)
Infinite AgentsModelling a community with a large number of agents. For instance, a continuum πΌ β [0,1] of agents
Network Effect Functionβ π is an increasing function that captures positive network effect on agents
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NETWORK EFFECT RECAP β SET UP
Homogeneous Utility FunctionEach agent has the utility function: π’ π π,π‘ , π₯π , ππ‘ , ππ‘β1 = π₯π πβ ππ‘β1 + 1 β π β ππ‘ β π π π,π‘, where π π,π‘ β {0,1}
denotes the strategy of agent π at time π‘, and ππ‘ denotes the fraction of population that choose π = 1
Type DistributionEach agent has a type π₯π, which describes his/her utility from taking action π = 1. π₯π comes from the
distribution πΊ(π₯) (CDF)
Infinite AgentsModelling a community with a large number of agents. For instance, a continuum πΌ β [0,1] of agents
Network Effect Functionβ π is an increasing function that captures positive network effects on agents
Other Constant Parametersπ: cost of taking action π = 1
π: weight of the network effect from the previous period
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NETWORK EFFECT RECAP β MAIN RESULT
Fixed Point Equation
Only agents that that will have positive utilities from taking action π = 1 will take this action. Recall that the utility
function has the form:
π’ π π,π‘ , π₯π , ππ‘ , ππ‘β1 = π₯π πβ ππ‘β1 + 1 β π β ππ‘ β π π π,π‘
Thus, only the agents with the following type will take action π = 1:
π₯π >π
πβ ππ‘β1 + 1 β π β ππ‘
Thus, we have the fixed point equation for the fraction of population that take action π = 1:
ππ‘ = 1 β πΊ(π
πβ ππ‘β1 + 1 β π β ππ‘)
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PROGRAM DEMO
03PART
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PROGRAM DEMO β STRUCTURE
Part 1:
Solve for equilibrium
Part 2-1:
Finite Agents Simulation
Part 2-2:
Analytical Form Analysis
User input 1: type distribution, network effect function, parametersβ¦
Output of equilibrium from an analysis using a finite number of agents
User input 2: input an initial state from which an equilibrium will be derived
Output of equilibrium from the closed form analysis based on the fixed point equation
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PROGRAM DEMO β ASSUMPTIONS
Type Distribution
πΊ π₯ =πΎ + π½
1 + π½
1 + π½ πβπΌβ²
π₯
πΎ + π½πβπΌβ²
π₯
Network Effect Function
πΊβ1 π =βπΌβ²
πππ(πΎπ
πΎ + π½ β π½π)
β π = π
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PROGRAM DEMO β ASSUMPTIONS
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PROGRAM DEMO
Demo in Spyder
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