Paradoxes in Social Networks with Multiple Products Krzysztof R. Apt - - PowerPoint PPT Presentation

Paradoxes in Social Networks with Multiple Products

Krzysztof R. Apt

CWI and University of Amsterdam

Joint work with Evangelos Markakis and Sunil Simon

Paradox of Choice (B. Schwartz, 2005)

[Gut Feelings, G. Gigerenzer, 2008]

The more options one has, the more possibilities for experiencing conflict arise, and the more difficult it becomes to compare the options. There is a point where more options, products, and choices hurt both seller and consumer.

Krzysztof R. Apt Paradoxes in Social Networks with Multiple Products

Speculative Bubbles

[Bubbles without Markets, R. Shiller, 2012]

A speculative bubble is a social epidemic whose contagion is mediated by price movements. News of price increase enriches the early investors. [...] The excitement then lures more and more people into the market, which causes prices to increase further, attracting yet more people. [...] After the bubble bursts, the same contagion fuels a precipitous collapse, as falling prices cause more and more people to exit the market.

Krzysztof R. Apt Paradoxes in Social Networks with Multiple Products

Plan

Objective: To understand such paradoxes and phenomena. Tools:

◮ a model of social networks, ◮ strategic games. Krzysztof R. Apt Paradoxes in Social Networks with Multiple Products

Social networks

Essential components of our model

Finite set of agents. Influence of “friends”. Finite product set for each agent. Cost of adopting a product.

Krzysztof R. Apt Paradoxes in Social Networks with Multiple Products

Social networks

Essential components of our model

Finite set of agents. Influence of “friends”. Finite product set for each agent. Cost of adopting a product. Example 4 1 3 2

0.4 0.5 0.3 0.6

Krzysztof R. Apt Paradoxes in Social Networks with Multiple Products

Social networks

Essential components of our model

Finite set of agents. Influence of “friends”. Finite product set for each agent. Cost of adopting a product. Example 4

{•}

1

{•, •}

3

{•, •}

2

{•, •} 0.4 0.5 0.3 0.6

Krzysztof R. Apt Paradoxes in Social Networks with Multiple Products

Social networks

Essential components of our model

Finite set of agents. Influence of “friends”. Finite product set for each agent. Cost of adopting a product. Example 4

{•} 0.5

1

0.3 {•, •}

3

{•, •} 0.2

2

{•, •} 0.4 0.4 0.5 0.3 0.6

Krzysztof R. Apt Paradoxes in Social Networks with Multiple Products

The model

Social network [Apt, Markakis 2011]

Weighted directed graph, 0 < wij ≤ 1: weight of the edge i → j. Products: A finite set of products P. Product assignment P: assigns to each agent a non-empty set of products. Cost function: θ(i, t) ∈ (0, 1], for each agent i and product t ∈ P(i).

Krzysztof R. Apt Paradoxes in Social Networks with Multiple Products

The associated strategic game

Interaction between agents: Each agent i can adopt a product from the set P(i) or choose not to adopt any product (t0).

Social network games

Players: Agents in the network. Strategies: Set of strategies for player i is P(i) ∪ {t0}. Payoff: Given a joint strategy s and an agent i,

Krzysztof R. Apt Paradoxes in Social Networks with Multiple Products

The associated strategic game

Interaction between agents: Each agent i can adopt a product from the set P(i) or choose not to adopt any product (t0).

Social network games

Players: Agents in the network. Strategies: Set of strategies for player i is P(i) ∪ {t0}. Payoff: Given a joint strategy s and an agent i,

◮ pi(s) =

   if si = t0

• j∈N

wji − θ(i, t) if si = t, for some t ∈ P(i) N: the set of neighbours of i who adopted in s the product t.

Krzysztof R. Apt Paradoxes in Social Networks with Multiple Products

Example

1

{•}

2

{•, •}

3

{•, •}

4 {•} 5

{•, •}

6

{•} 0.1 0.1 0.1 0.2 0.1 0.1 0.2 0.1 0.1 0.2

Cost is 0.01 for all the players and all the products. P = {•, •, •}

Krzysztof R. Apt Paradoxes in Social Networks with Multiple Products

Example

1

{•}

2

{•, •}

3

{•, •}

4 {•} 5

{•, •}

6

{•} 0.1 0.1 0.1 0.2 0.1 0.1 0.2 0.1 0.1 0.2

Cost is 0.01 for all the players and all the products. P = {•, •, •} Payoff: p1(s) = 0

Krzysztof R. Apt Paradoxes in Social Networks with Multiple Products

Example

1

{•}

2

{•, •}

3

{•, •}

4 {•} 5

{•, •}

6

{•} 0.1 0.1 0.1 0.2 0.1 0.1 0.2 0.1 0.1 0.2

Cost is 0.01 for all the players and all the products. P = {•, •, •} Payoff: p1(s) = 0 p2(s) = −0.01

Krzysztof R. Apt Paradoxes in Social Networks with Multiple Products

Example

1

{•}

2

{•, •}

3

{•, •}

4 {•} 5

{•, •}

6

{•} 0.1 0.1 0.1 0.2 0.1 0.1 0.2 0.1 0.1 0.2

Cost is 0.01 for all the players and all the products. P = {•, •, •} Payoff: p1(s) = 0 p2(s) = −0.01 p3(s) = p4(s) = p5(s) = p6(s) = 0.1 − 0.01 = 0.09

Krzysztof R. Apt Paradoxes in Social Networks with Multiple Products

Social network games

Properties

Graphical game: The payoff for each player depends only on the choices made by his neighbours. Join the crowd property: The payoff of each player weakly increases if more players choose the same strategy.

Krzysztof R. Apt Paradoxes in Social Networks with Multiple Products

Solution concept – Nash equilibrium

Notation

s: a joint strategy, also written as (si, s−i). s−i: the joint strategy of the opponents of player i.

Best response

A strategy si of player i is a best response to s−i if for all s′

i,

pi(s′

i , s−i) ≤ pi(si, s−i).

Nash equilibrium

A strategy profile s is a Nash equilibrium if for all players i, si is a best response to s−i.

Krzysztof R. Apt Paradoxes in Social Networks with Multiple Products

Example

1

{•}

2

{•, •}

3

{•, •}

4 {•} 5

{•, •}

6

{•} 0.1 0.1 0.1 0.2 0.1 0.1 0.2 0.1 0.1 0.2

Cost is 0.01 for all the players. P = {•, •, •} This is not a Nash equilibrium. Payoff: p1(s) = 0 p2(s) = −0.01 p3(s) = p4(s) = p5(s) = p6(s) = 0.1 − 0.01 = 0.09

Krzysztof R. Apt Paradoxes in Social Networks with Multiple Products

Example

1

{•}

2

{•, •}

3

{•, •}

4 {•} 5

{•, •}

6

{•} 0.1 0.1 0.1 0.2 0.1 0.1 0.2 0.1 0.1 0.2

Cost is 0.01 for all the players. P = {•, •, •} This is a Nash equilibrium. Payoff: p1(s) = 0.09 p2(s) = 0.09 p3(s) = p4(s) = p5(s) = p6(s) = 0.1 − 0.01 = 0.09

Krzysztof R. Apt Paradoxes in Social Networks with Multiple Products

Paradox 1

Adding a product to a social network can trigger a sequence of changes that will lead the agents from one Nash equilibrium to a new one that is worse for everybody.

Krzysztof R. Apt Paradoxes in Social Networks with Multiple Products

Example

1

{•}

2

{•, •}

3

{•, •}

4 {•} 5

{•, •}

6

{•} 0.1 0.1 0.1 0.2 0.1 0.1 0.2 0.1 0.1 0.2

Cost θ is constant, 0 < θ < 0.1.

Krzysztof R. Apt Paradoxes in Social Networks with Multiple Products

Example

1

{•}

2

{•, •}

3

{•, •}

4 {•} 5

{•, •}

6

{•} 0.1 0.1 0.1 0.2 0.1 0.1 0.2 0.1 0.1 0.2

Cost θ is constant, 0 < θ < 0.1. This is a Nash equilibrium. The payoff to each player is 0.1 − θ > 0.

Krzysztof R. Apt Paradoxes in Social Networks with Multiple Products

Example

1

{•}

2

{•, •}

3

{•, •}

4 {•}

• 5

{•, •}

6

{•} 0.1 0.1 0.1 0.2 0.1 0.1 0.2 0.1 0.1 0.2

Cost θ is constant, 0 < θ < 0.1. This is not a Nash equilibrium.

Krzysztof R. Apt Paradoxes in Social Networks with Multiple Products

Example

1

{•}

2

{•, •}

3

{•, •}

4 {•}

• 5

{•, •}

6

{•} 0.1 0.1 0.1 0.2 0.1 0.1 0.2 0.1 0.1 0.2

Cost θ is constant, 0 < θ < 0.1. This is not a Nash equilibrium.

Krzysztof R. Apt Paradoxes in Social Networks with Multiple Products

Example

1

{•}

2

{•, •}

3

{•, •}

4 {•}

• 5

{•, •}

6

{•} 0.1 0.1 0.1 0.2 0.1 0.1 0.2 0.1 0.1 0.2

Cost θ is constant, 0 < θ < 0.1. This is not a Nash equilibrium.

Krzysztof R. Apt Paradoxes in Social Networks with Multiple Products

Example

1

{•}

2

{•, •}

3

{•, •}

4 {•}

• 5

{•, •}

6

{•} 0.1 0.1 0.1 0.2 0.1 0.1 0.2 0.1 0.1 0.2

Cost θ is constant, 0 < θ < 0.1. This is not a Nash equilibrium.

Krzysztof R. Apt Paradoxes in Social Networks with Multiple Products

Example

1

{•}

2

{•, •}

3

{•, •}

4 {•}

• 5

{•, •}

6

{•} 0.1 0.1 0.1 0.2 0.1 0.1 0.2 0.1 0.1 0.2

Cost θ is constant, 0 < θ < 0.1. This is not a Nash equilibrium.

Krzysztof R. Apt Paradoxes in Social Networks with Multiple Products

Example

1

{•}

2

{•, •}

3

{•, •}

4 {•}

• 5

{•, •}

6

{•} 0.1 0.1 0.1 0.2 0.1 0.1 0.2 0.1 0.1 0.2

Cost θ is constant, 0 < θ < 0.1. This is not a Nash equilibrium.

Krzysztof R. Apt Paradoxes in Social Networks with Multiple Products

Example

1

{•}

2

{•, •}

3

{•, •}

4 {•}

• 5

{•, •}

6

{•} 0.1 0.1 0.1 0.2 0.1 0.1 0.2 0.1 0.1 0.2

Cost θ is constant, 0 < θ < 0.1. This is not a Nash equilibrium.

Krzysztof R. Apt Paradoxes in Social Networks with Multiple Products

Example

1

{•}

2

{•, •}

3

{•, •}

4 {•}

• 5

{•, •}

6

{•} 0.1 0.1 0.1 0.2 0.1 0.1 0.2 0.1 0.1 0.2

Cost θ is constant, 0 < θ < 0.1. This is not a Nash equilibrium.

Krzysztof R. Apt Paradoxes in Social Networks with Multiple Products

Example

1

{•}

2

{•, •}

3

{•, •}

4 {•}

• 5

{•, •}

6

{•} 0.1 0.1 0.1 0.2 0.1 0.1 0.2 0.1 0.1 0.2

Cost θ is constant, 0 < θ < 0.1. This is not a Nash equilibrium.

Krzysztof R. Apt Paradoxes in Social Networks with Multiple Products

Example

1

{•}

2

{•, •}

3

{•, •}

4 {•}

• 5

{•, •}

6

{•} 0.1 0.1 0.1 0.2 0.1 0.1 0.2 0.1 0.1 0.2

Cost θ is constant, 0 < θ < 0.1. This is not a Nash equilibrium.

Krzysztof R. Apt Paradoxes in Social Networks with Multiple Products

Example

1

{•}

2

{•, •}

3

{•, •}

4 {•}

• 5

{•, •}

6

{•} 0.1 0.1 0.1 0.2 0.1 0.1 0.2 0.1 0.1 0.2

Cost θ is constant, 0 < θ < 0.1. This is a Nash equilibrium. The payoff to each player is 0.

Krzysztof R. Apt Paradoxes in Social Networks with Multiple Products

Paradox 2

Removing a product from a social network can result in a sequence of changes that will lead the agents from one Nash equilibrium to a new one that is better for everybody.

Krzysztof R. Apt Paradoxes in Social Networks with Multiple Products

Example

1

{•}

2

{•}

3

{•, •}

4

{•, •} w w w w w w w w

Cost θ is product independent. The weight of each edge is w, where w > θ. Note Each node has two incoming edges.

Krzysztof R. Apt Paradoxes in Social Networks with Multiple Products

Example

1

{•}

2

{•}

3

{•, •}

4

{•, •} w w w w w w w w

Cost θ is product independent. The weight of each edge is w, where w > θ. This is a Nash equilibrium. The payoff to each player is w − θ.

Krzysztof R. Apt Paradoxes in Social Networks with Multiple Products

Example

1

{•}

2

{•}

3

{•}

4

{•, •} w w w w w w w w

Cost θ is product independent. The weight of each edge is w, where w > θ. This is not a legal joint strategy.

Krzysztof R. Apt Paradoxes in Social Networks with Multiple Products

Example

1

{•}

2

{•}

3

{•}

4

{•, •} w w w w w w w w

Cost θ is product independent. The weight of each edge is w, where w > θ. This is not a Nash equilibrium.

Krzysztof R. Apt Paradoxes in Social Networks with Multiple Products

Example

1

{•}

2

{•}

3

{•}

4

{•, •} w w w w w w w w

Cost θ is product independent. The weight of each edge is w, where w > θ. This is not a Nash equilibrium.

Krzysztof R. Apt Paradoxes in Social Networks with Multiple Products

Example

1

{•}

2

{•}

3

{•}

4

{•, •} w w w w w w w w

Cost θ is product independent. The weight of each edge is w, where w > θ. This is not a Nash equilibrium.

Krzysztof R. Apt Paradoxes in Social Networks with Multiple Products

Example

1

{•}

2

{•}

3

{•}

4

{•, •} w w w w w w w w

Cost θ is product independent. The weight of each edge is w, where w > θ. This is a Nash equilibrium. The payoff to each player is 2w − θ.

Krzysztof R. Apt Paradoxes in Social Networks with Multiple Products

When the paradoxes cannot occur

Theorem Paradox 1 cannot arise when the graph has no source nodes, each Nash equilibrium uses at most one product.

Krzysztof R. Apt Paradoxes in Social Networks with Multiple Products

Final remarks

Open problem: Does a social network exist that exhibits paradox 1 for every triggered sequence of changes? Alternative approach: Obligatory product selection (no t0). In this setup the above problem has an affirmative answer.

Krzysztof R. Apt Paradoxes in Social Networks with Multiple Products

Conclusions

A simple model introduced that allowed us to explain well-known paradoxes and phenomena. Needed: Identify other conditions that guarantee that these paradoxes cannot arise. Future work: Study the effects of the price fluctuations.

Krzysztof R. Apt Paradoxes in Social Networks with Multiple Products

Thank you

Krzysztof R. Apt Paradoxes in Social Networks with Multiple Products