# Game Theory - Applications in real life by pradeeban

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```									    Game Theory

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Agenda

Games

Evolutionary Game Theory

Modeling Network Traffic using Game
Theory

Auctions

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Evolutionary Game Theory

Applying Game Theory for Evolutionary
Biology.

Fitness as a result of interaction.

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The Body-Size Game
Beetle2

Small       Large

Beetle1    Small   5, 5        1, 8

Large   8, 1        3, 3

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Evolutionary Stable Strategies
S      L

S     5, 5   1, 8

L     8, 1   3, 3

Expected payoff:
To a small beetle during a random
encounter =
5 * probability of meeting another small +
1 * probability of meeting a large =
5 (1 – x ) + 1. x = 5 – 4x.
5
Evolutionary Stable Strategies
S      L

S     5, 5   1, 8

L     8, 1   3, 3

Expected payoff:
To a large beetle during a random
encounter =
8 * probability of meeting a small +
3 * probability of meeting another large =
8 (1 – x ) + 3. x = 8 – 5x.
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Evolutionary Stable Strategies
S      L

S      5, 5   1, 8

L      8, 1   3, 3

For a small enough fraction of x,
• 8 – 5 x > 5 – 4 x.

“Small” is not evolutionarily stable.

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Evolutionary Stable Strategies
S      L

S     5, 5   1, 8

L     8, 1   3, 3

Expected payoff:
To a small beetle = 5.x + 1.(1 – x) = 1 + 4x

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Evolutionary Stable Strategies
S      L

S     5, 5   1, 8

L     8, 1   3, 3

Expected payoff:
To a small beetle = 5.x + 1.(1 – x) = 1 + 4x
To a large beetle = 8.x + 3.(1 – x) = 3 + 5x

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Evolutionary Stable Strategies
S      L

S     5, 5   1, 8

L     8, 1   3, 3

Expected payoff:
To a small beetle = 5.x + 1.(1 – x) = 1 + 4x
To a large beetle = 8.x + 3.(1 – x) = 3 + 5x
3 + 5x > 1 + 4x
“Large” is evolutionarily stable.
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Is S Evolutionarily Stable?
S      T

S   a, a   b, c

T   c, b   d, d

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Is S Evolutionarily Stable?
S      T


a (1 - x) + b.x >
S   a, a   b, c
c (1 – x) + d.x
T   c, b   d, d


a > c or a = c & b > d

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Is S Evolutionarily Stable?
S      T


a (1 - x) + b.x >
S    a, a   b, c
c (1 – x) + d.x
T    c, b   d, d


a > c or a = c & b > d


S, S – best response to the other player
– Evolutionarily Stable (ES) → Nash
Equilibrium.
– Strict Nash Equilibrium (unique
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response to the other player) → ES
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Modeling Network Traffic

Game-theoretic reasoning
– Traveling through a
transportation network.
– Sending packets through the
Internet.

Evaluate routes in presence of the
congestion resulting from the decisions
of myself and others.

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Traffic at Equilibrium:
4000 Cars, A → B


Time taken at equilibrium for every driver =

Equally divided between ACB and ADB =
2000/100 + 45 = 65
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Nash Equilibrium

Equal Balance → Nash Equilibrium.

Nash Equilibrium → Equal Balance.

No incentive to switch over to the other
route.

Any unequal weights will not be in Nash
Equilibrium.

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
Adding capacity to a network can
sometimes actually slow down the traffic.

network can sometimes hurt performance at
equilibrium.

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4000 Cars, A → B


The unique Nash equilibrium leads towards
using ACDB.

Time taken for every driver = 4000/100 + 0
19       + 4000/100 = 80
4000 Cars, A → B


No benefit by changing their route, for an
individual.

Changing will increase the travel time as,
20       45 + 4000/100 = 85
Case Study:

Destruction of a six-lane highway to build a
public park actually improved travel time
into and out of the city in Seoul, South
Korea.

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Auction

Ascending-Bid Auctions (English Auctions)

Descending-Bid Auctions (Dutch Auctions)

First-Price Sealed-Bid Auctions

Second-Price Sealed-Bid Auctions

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Ascending-Bid Auction

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10 \$

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30 \$

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50 \$

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70 \$

Awarded for: 70 \$ !!!

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Descending-Bid Auction

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100 \$

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90 \$

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80 \$

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70 \$

Awarded for: 70 \$ !!!

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First-Price Sealed-Bid Auction

Bidders submit simultaneous sealed-bids to
the seller.

Seller opens the sealed bids together.

Highest bid wins the object.

Pays the value of his bid.

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Second-Price Sealed-Bid Auction

Bidders submit simultaneous sealed-bids to
the seller.

Seller opens the sealed bids together.

Highest bid wins the object.

Pays the price of the second highest bid.

If multiple bids share the highest position?

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Second-Price Sealed-Bid Auction

Bidders submit simultaneous sealed-bids to
the seller.

Seller opens the sealed bids together.

Highest bid wins the object.

Pays the price of the second highest bid.

If multiple bids share the highest position
– one of the bidders awarded the
object.
– for the same price.

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First-Price Sealed-Bid Auction

Similar to descending bid auction.

Takes the risk, without knowing the value
of others.

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Second-Price Sealed-Bid Auction

Similar to ascending bid auction.

Others' price is known.

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Payoff and true value

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When Auctions?

When the true-value of the buyer is
unknown.

Bidding one's true value is the dominant
strategy. (Really?)

Reselling?
– Winner's curse.

All-pay Auctions?

39
References
Easley, David., Kleinberg, Jon, “Networks,
Crowds, and Markets: Reasoning about a
Highly Connected World,” Cambridge
University Press, 2010: 155 – 268.

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Questions?

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Thank you!

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