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Social Media Games
Richard Southwell
NCEL Workshop
May 2013
Social Media Game
Players can decide whether to spend their time generating new content, or reading/following the content produced by other users
Player 2 spends all their time following user 3
Player 3 spends all their time creating new content
Player p chooses strategy (t1p, t2
p,.., tnp)
tk
p ≥ 0 is the amount of time p spends following k ≠ p tp
p ≥ 0 is the amount of time p spends generating new content t1
p+ t2p+..+ tn
p =Tp
amount of time p has
amount of time p has
amount p gains from reading k’s content amount p gains from k reading their content
Player p chooses strategy (t1p, t2
p,.., tnp)
tk
p ≥ 0 is the amount of time p spends following k ≠ p tp
p ≥ 0 is the amount of time p spends generating new content t1
p+ t2p+..+ tn
p =Tp [Payoff to p] = ∑ k ≠ p fread
p,k (tkp, tk
k) + ∑ k ≠ p fwritep,k (tp
k, tpp)
Functions fread
p,k (x, y) and fwritep,k(x,y) are increasing in x and y
amount of time p has
amount p gains from reading k’s content amount p gains from k reading their content
tkp tk
k
freadp,k
tpk tp
p
fwritep,k
Player p chooses strategy (t1p, t2
p,.., tnp)
tk
p ≥ 0 is the amount of time p spends following k ≠ p tp
p ≥ 0 is the amount of time p spends generating new content t1
p+ t2p+..+ tn
p =Tp [Payoff to p] = ∑ k ≠ p fread
p,k (tkp, tk
k) + ∑ k ≠ p fwritep,k (tp
k, tpp)
Functions fread
p,k (x, y) and fwritep,k(x,y) are increasing in x and y
Also fread
p,k(r,0)= freadp,k(0,r)= fwrite
p,k(r,0)=fwritep,k =0 for all r
T1=T2=..=Tn=1 fread
p,k (tkp, tk
k)=r tkp tk
k
fwritep,k (tp
k, tpp)=w tp
k tpp
[Payoff to p] = r ∑ k ≠ p tkp tk
k +w ∑ k ≠ p tpk tp
p
Homogenous Linear Case
T1=T2=..=Tn=1 fread
p,k (tkp, tk
k)=r tkp tk
k
fwritep,k (tp
k, tpp)=w tp
k tpp
[Payoff to p] = r ∑ k ≠ p tkp tk
k +w ∑ k ≠ p tpk tp
p
Homogenous Linear Case constant representing how much players like reading content
constant representing how much players like having their content read
fraction of k’s time they spend following p
fraction of p’s time they spend generating content
(t11, t2
1)=(0,1) (t12, t2
2)=(0,1)
T1=T2=..=Tn=1 fread
p,k (tkp, tk
k)=r tkp tk
k
fwritep,k (tp
k, tpp)=w tp
k tpp
[Payoff to p] = r ∑ k ≠ p tkp tk
k +w ∑ k ≠ p tpk tp
p
Homogenous Linear Case constant representing how much players like reading content
constant representing how much players like having their content read
fraction of k’s time they spend following p
fraction of p’s time they spend generating content
(t11, t2
1)=(0,1) (t12, t2
2)=(0,1) Theorem (t1,t2,..,tn) is a pure Nash equilibrium if and only if either [one player is generating content and all other players are following them] or [each player generating content has at least r/w followers]
T1=T2=..=Tn=1 fread
p,k (tkp, tk
k)=r tkp tk
k
fwritep,k (tp
k, tpp)=w tp
k tpp
[Payoff to p] = r ∑ k ≠ p tkp tk
k +w ∑ k ≠ p tpk tp
p
Homogenous Linear Case constant representing how much players like reading content
constant representing how much players like having their content read
fraction of k’s time they spend following p
fraction of p’s time they spend generating content
(t11, t2
1)=(0,1) (t12, t2
2)=(0,1) Theorem (t1,t2,..,tn) is a pure Nash equilibrium if and only if either [one player is generating content and all other players are following them] or [each player generating content has at least r/w followers]
Fundamentally different pure Nash equilibria when n=9, r=3, w=2
freadp,k (tk
p, tkk)=Rp,k tk
p tkk
fwritep,k (tp
k, tpp)= Wp,k tp
k tpp
[Payoff to p] =∑ k ≠ p Rp,k tkp tk
k + ∑ k ≠ p Wp,k tpk tp
p
General Linear Case
freadp,k (tk
p, tkk)=Rp,k tk
p tkk
fwritep,k (tp
k, tpp)= Wp,k tp
k tpp
[Payoff to p] =∑ k ≠ p Rp,k tkp tk
k + ∑ k ≠ p Wp,k tpk tp
p
General Linear Case
p k
Rp,k measures how much p likes reading k’s content
Wp,k measures how much p likes it when k reads their content
freadp,k (tk
p, tkk)=Rp,k tk
p tkk
fwritep,k (tp
k, tpp)= Wp,k tp
k tpp
[Payoff to p] =∑ k ≠ p Rp,k tkp tk
k + ∑ k ≠ p Wp,k tpk tp
p
General Linear Case
p k
Rp,k measures how much p likes reading k’s content
Wp,k measures how much p likes it when k reads their content
Theorem If we keep letting the players do best response updates asynchronously, the system will eventually reach a pure Nash equilibrium
freadp,k (tk
p, tkk)=Rp,k tk
k min{tkp, αk
p}
fwritep,k (tp
k, tpp)= Wp,k tp
k min{tpk, αp
k}
[Payoff to p] =∑ k ≠ p Rp,k tk
k min{tkp, αk
p} + ∑ k ≠ p Wp,k tpp min{tp
k, αpk}
Constrained Linear Case
freadp,k (tk
p, tkk)=Rp,k tk
k min{tkp, αk
p}
fwritep,k (tp
k, tpp)= Wp,k tp
k min{tpk, αp
k}
[Payoff to p] =∑ k ≠ p Rp,k tk
k min{tkp, αk
p} + ∑ k ≠ p Wp,k tpp min{tp
k, αpk}
Constrained Linear Case
tkp
freadp,k
αkp
freadp,k (tk
p, tkk)=Rp,k tk
k min{tkp, αk
p}
fwritep,k (tp
k, tpp)= Wp,k tp
k min{tpk, αp
k}
[Payoff to p] =∑ k ≠ p Rp,k tk
k min{tkp, αk
p} + ∑ k ≠ p Wp,k tpp min{tp
k, αpk}
Constrained Linear Case
tkp
freadp,k
αkp
Theorem In the symmetric case when Wp,k=Rp,k for every p and k, the function V(t) = 0.5 ∑ p [payoff to p in t] is an exact potential function
freadp,k (tk
p, tkk)=Rp,k tk
k min{tkp, αk
p}
fwritep,k (tp
k, tpp)= Wp,k tp
k min{tpk, αp
k}
[Payoff to p] =∑ k ≠ p Rp,k tk
k min{tkp, αk
p} + ∑ k ≠ p Wp,k tpp min{tp
k, αpk}
Constrained Linear Case
tkp
freadp,k
αkp
Theorem In the symmetric case when Wp,k=Rp,k for every p and k, the function V(t) = 0.5 ∑ p [payoff to p in t] is an exact potential function
better response updates converge to pure Nash equilibria
The Nash equilibria are the local maxima of the social welfare
tkp tk
k
freadp,k
tpk tp
p
fwritep,k
[Payoff to p] = ∑ k ≠ p fread
p,k (tkp, tk
k) + ∑ k ≠ p fwritep,k (tp
k, tpp)
Functions fread
p,k (x, y) and fwritep,k(x,y) are increasing in x and y
Also fread
p,k(r,0)= freadp,k(0,r)= fwrite
p,k(r,0)=fwritep,k =0 for all r
General Case
Theorem A strategy profile where one player spends all their time generating content, and all the other players spend all their time watching that player will always be a pure Nash equilibrium.