A presentation on game theory in social networking and routing network traffic using game theory in a weighted directed graph

1. CIS: 5524
Routing Network Traffic
Using Game Theory
Nimit Johri
nimit.johri@temple.edu

2. What is a game?

3. A game is an interaction
between decision makers
where the happiness of
each participant with the
outcome depends not just
on his or her own decisions
but on the decisions made
by everyone!

4. Lets play a game!

5. Suppose there are two winter clothing firms, Blue
and Grey, who are strategizing to maximize their
sales in the next winter
• They can either decide to produce low-priced products
or up-scaled ones
• 60% of the population buys low-prized products and
40% buys up-scaled ones
80%
20%

6. Which product should Blue firm produce?
Up-Scale
Low-priced

7. Not a fun game!

8. An easier game first – Prisoner’s Dilemma
The police have apprehended two suspects of robbery at McDonalds at Broad Street
but there is not enough evidence to convict either of them
However they both resisted arrest and can be charged with a lesser crime, which
would carry a one year sentence
Each of them was told the following
• If you confess and your partner doesn’t then you will be released and your
partner will be charged for the robbery, carrying a sentence of ten years
• If you both confess then you both will be charged but the sentence will be only
of 4 years because of guilty plea
• If neither of you confesses then you both will be charged for resisting arrest

9. From suspect 1’s perspective, what should we do?
We can make either of the two choices
• Confess
• Not Confess
Let us summarize the information we have
Not Confess Confess
Not Confess
Confess
Suspect 2
Suspect 1
-1, -1 -10, 0
0, -10 4, 4

10. From suspect 1’s perspective, what should we do?
Not Confess Confess
Not Confess -1, -1 -10, 0
Confess 0, -10 -4, -4
Suspect 2
Suspect 1
Lets analyze both options:
• If the other suspect Confesses, then we get 10 years by not confessing and 4
years by confessing, so in this case we should Confess
• If the other suspect does Not Confesses, then we get 1 year by not confessing
and 0 years by confessing, so in this case too we should Confess

11. Strictly Dominant Strategy
We can conclude that its always best to confess regardless of what the other player
chooses, we should always confess
• A strategy which is the best choice regardless of what the other player
chooses
• Based on the given conditions, strictly dominant strategy may be available for
all the players, a subset of players or none of the players
This is known as Strictly Dominant Strategy

12. Three Client Game
Suppose there are two consulting firms and each wants to do business with one of the
three clients A, B & C but can only approach exactly one of them
The results of their two decisions will work out as follows:
• If they both approach same client then each gets half of the business
• Firm 1 is too small to attract business on it’s own so if they both choose different
clients then Firm 1 gets 0 payoff
• If Firm 2 approach clients B or C on it’s own, it will their full business but client A is
large and will only do business if approached by both
• As client A is larger it’s payoff is 8 while for the other two it is 2

13. Three Client Game
Let us summarize in the payoff matrix
A B C
A 4, 4 0, 2 0, 2
B 0, 0 1, 1 0, 2
C 0, 0 0, 2 1, 1
Firm 2
Firm 1
There is no dominant strategy available for any firm in the above payoff matrix as:
• For firm 1, the best response always depends on what firm 2 is selecting which can
be mapped as A-A, B-B, C-C
• For firm 2 also the best response always depends on what firm 1 is selecting which
can be mapped as A-A, B-C, C-B

14. Nash Equilibrium
A B C
A 4, 4 0, 2 0, 2
B 0, 0 1, 1 0, 2
C 0, 0 0, 2 1, 1
Firm 2
Firm 1
• Even when there are no dominant strategies, we should expect players to use
strategies which are best responses to each other
• The idea is that if players chooses strategies which are best responses to each
other, then no player has an incentive to deviate to an alternate strategy
• Therefore we can say that the system is in a state of equilibrium even with the
absence of a dominant strategy.
• This is known as Nash Equilibrium

15. Modeling Network Traffic using
Game Theorey

16. Traffic at Equilibrium
• We can equate data packets traveling in a network of routers to highway system where
cars are traveling on highways from one place to another
• The above is a weighted directed graph where the edge weight is the travel time for that
edge
• We consider the cost for travelling from node A to B

17. • The edges AC and DB are sensitive to congestion as their cost depends is a function of x
where x is the number of cars on that edge
• The edges AD and CB are insensitive to congestion
• Suppose 4000 cars need to travel from A to B
• If all cars take Route through C or through D then travel cost is 45 + 40 = 85
• If the cars divide evenly across both the routes then travel cost reduces to 45 + 20 = 65

18. • There is no dominant strategy as any driver could choose either of the route
• But we can observe a Nash Equilibrium when the cars evenly divide across the two routes
• If there are uneven number of cars on the routes then one route will have smaller travel
cost and the drivers in the larger cost route will have the incentive to switch their current
route
• If the cars divide evenly then the travel cost is same on both the routes then no driver
has an incentive to switch their current route
• Hence the system is in a state of equilibrium

19. The more, the better?

20. Braess’s Paradox
• Suppose the city government decides to build a super fast highway from C to D
• We model this travel time as 0
• Now, there is a unique Nash equilibrium in this new network which leads to
worse travel time for everyone
• We call this Braess’s paradox

21. Braess’s Paradox
• Now every driver will follow route AC – CD – DB
• The system is in equilibrium as no driver has an incentive in switching to any
other route
• Travel time for any other route is 45 + 40 = 85
• The travel time for this route is 40 + 0 + 40 = 80 while earlier it was 65

22. But why does this happen?

23. Braess’s Paradox
• This happens because the creation of edge CD made route through CD a
dominant strategy for all drivers
• Regardless of the traffic pattern, every driver gain by switching to the route
through CD as it acts like a vortex that draws drivers into it
• Therefore Braess’s paradox states that addition of resources to a network does
not always results in an increased performance

24. HomeWork
Suppose you made it to the final of the Pokemon championship and the final match is one vs
one. You have a Fire type (Charizard) and a Water type (Blastoise) and your opponent has the
same
The results of the match can go as follows:
• If Fire type is against Water type then Water type wins
• If Water type is against a Water type then it’s a draw and both get half the glory
• If Fire type is against Fire type then they fight fiercely ending up critically injuring each
other, enough for the tournament to be called off and no one wins
The possibilities have been summarized in the below payoff table
Fire Water
Fire 0, 0 0, 3
Water 3, 0 2, 2
Which Pokemon would you choose?
Fire Type (Charizard) Water Type (Blastoise)

25. Extra Slides

26. Suppose there are two winter clothing firms, Blue
and Grey, who are strategizing to maximize their
profits in the next winter
• They can either decide to produce low-priced products
or up-scaled ones
• 60% of the population buys low-prized products and
40% buys up-scaled ones
80%
20%

27. Remember, the objective is to maximize sales
Let us explore all possible outcomes
• If both firms market to different segments then Blue
gets payoff 0.60 and Grey gets payoff 0.40
• If both firms market low-prized segment then Blue gets
80% of 60 that is payoff 0.48 while Grey gets 20% which
is a payoff 0.12
• If both firms market upscale segment then Blue gets
payoff 32 while Grey gets payoff 0.08

28. Remember, the objective is to maximize sales
Let us summarize this information in a payoff table
Low-prized Upscale
Low-prized .48, .02 .60, .40
Upscale .40, .60 .32, .08
Grey
Blue
• Clearly Blue has a strictly dominant strategy by marketing low-prize
• Now, Grey can predict that Blue will play according to its dominant
strategy and can respond with its best response which is marketing
upscale in this case
• Hence when only one player had the dominant strategy, still the
outcome of the game can be predicted

29. Battle of the Sexes
Suppose a couple is planning a date night and wants to watch a movie together
where the wife wants to watch a romantic movie (Five Feet Apart) while the
husband wants to watch an action movie (End Game)
Their date night can go as follows
• If they both decide to watch an action movie then both of then will enjoy but
the husband would be much happier
• If they both decide to watch a romantic movie then both of them will enjoy
but the wife would be much happier
• They both don’t want to sit In different theaters watching different movies on
their date night

30. Battle of the Sexes
Suppose a couple is planning a date night and wants to watch a movie together
where the wife wants to watch a romantic movie (Five Feet Apart) while the
husband wants to watch an action movie (End Game)
Let us summarize in the payoff matrix
Action Romantic
Action 2, 1 0, 0
Romantic 0, 0 1, 2
Wife
Husband
• There still exists an equilibrium as for a given choice, no player has an
incentive to switch to other strategy as the other leads to payoff 0
• But its hard to predict which equilibria will be played in this case as one gives
higher payoff to wife and the other gives higher payoff to husband