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.