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Дни науки в Skyeng:Теория Игр - математика конфликтов

GAME THEORY

What it is about?

Decision making in conflict and cooperation, including conflict of interests

Decision-makers are rational and intelligent

VOCABULAR & STRUCTURE

• Players, which are finitely many,

• Choose strategies, finitely many for each player,

• Got payoffs, dependent of the choice of strategies

• The result can be given as table (normal form)

I, II LEFT strategy RIGHT strategy

UP strategy Payoff, Payoff Payoff, Payoff

DOWN strategy Payoff, Payoff Payoff, Payoff

PRISONERS DILEMMA

GULITY NOT GULITY

GULITY -5, -5 0, -10

NOT GULITY -10, 0 -1, -1

NASH EQUILIBRIUM

A CHOICE OF STRATEGIES

SUCH THAT NO PLAYER

INCREASES HIS/HER PAYOFF

WHEN CHANGING HIS/HER STRATEGY

WHILE OTHERS KEEP THEIR

FRIEND OR FANCY

FRIEND FANCY

FANCY 4, 3 1, 1

FRIEND 0, 0 3, 4

MIXED EQUILIBRIUM

Assume that Player I chooses their fancy with frequency p

Assume that Player II chooses their fancy with frequency q

Payoff = 4p(1-q) + pq + 3(1-p)q = p(4 – 4q + q – 3q) +3q = p(4-6q) + 3q

Payoff = q(4-6p) + 3p

ANALYSIS

q>⅔ q<⅔

p>⅔ ↓, ↓ ⅔, ⅔ ↑, ↓ 1, 0

p<⅔ ↓, ↑ 0, 1 ↑, ↑ ⅔, ⅔

ASYMMETRIC GAME: ULTIMATUM

ACCEPT REJECT

FAIR 0.5, 0.5 0, 0

UNFAIR 0.8, 0.2 0, 0

ULTIMATUM: OTHER APPROACH

Let p be the share Player I proposes to keep for himself

Let q the maximal share Player II is ready to give up

Payoff = p, (1-p) if p ≤ q and 0, 0 otherwise

Equilibriums: p=q and p=1, q=0.

ZERO-SUM GAME: STRONGHOLD

LIGHT COLUMN HEAVY COLUMN

LIGHT COLUMN 0.1, -0.1 0.6, -0.6

HEAVY COLUMN

1, -1 0.5, -0.5

MIXED EQUILIBRIUM

Assume that Player I and Player II choose light column

with frequencies p, q

Payoff = 0.1pq + 0.6p(1-q) + (1-p)q + 0.5 (1-p) (1-q)

= p(0.1q + 0.6(1–q) - q – 0.5(1-q) ) +q + 0.5(1-q)

= p(0.1-q) + 0.5(q +1)

Payoff = - p(0.1-q) - 0.5(q +1) = q(p – 0.5) – 0.1p – 0.5

ANALYSIS

q>0.1 q<0.1

p>0.5 ↓, ↑ 0.5, 1 ↑, ↑ 1, 0.1

p<0.5 ↓, ↓ 0, 0.1 ↑, ↓ 0.5, 0

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