Wstp do Teorii Gier Zeus and Athena Zeus

Wstęp do Teorii Gier

Zeus and Athena • Zeus Music is a market leader in producing a modern audio equipment. • Athena Acoustics is a smaller but very innovative firm • Both firms have invented a new hexaphonic sound system of a total audio surrounding. (You hang a guy at some height and put 6 speakers around him) • There is uncertainty about the size of a market. – There is 50 -50 chance of small ($24 mln profits) and big ($40 mln profits) market • The two firms have to decide whether to launch a highest quality audio system designed for a very demanding customers or a cheaper system designed for less demanding customers.

Zeus and Athena • If the market is small, it is better to sell highest quality audio system • If the market is big, there will be more demand for lower quality system • Athena is prepared better than Zeus to produce the more innovative system • But Zeus is a widely known company and has more marketing potential in selling the less innovative system • Taking into account all these factors, analysts from Zeus estimated market shares of both firms • Since estimation was based on publicly available information Zeus believes that analysts from Athena have similar predictions.

Information sets to be added later

Different Assumptions about information A 1 • Both firms do not reveal their decisions about production (and they do not know what nature did)

Different Assumptions about information A 1 ATHENA LQ ZEUS HQ LQ (23, 9) (18, 14) HQ (18, 14) (20, 12)

Different Assumptions about information A 1 ATHENA LQ ZEUS HQ LQ (23, 9) (18, 14) HQ (18, 14) (20, 12) Mixed strategy Nash Equilibrium: for both firms the same: 2/7 LQ, 5/7 HQ Value of a game: (19 and 3/7 for Zeus; 12 and 4/7 for Athena)

Different Assumptions about information A 2 • Zeus has to make production decision earlier. • Athena – a smaller and more flexible firm – may decide after observing decision made by Zeus

Different Assumptions about information A 2 ATHENA ZEUS LQ HQ LQ/LQ (23, 9) (18, 14) LQ/HQ (23, 9) (20, 12) HQ/LQ (18, 14) HQ/HQ (18, 14) (20, 12) ATHENA ZEUS LQ HQ LQ/LQ (23, 9) (18, 14) LQ/HQ (23, 9) (20, 12)

Different Assumptions about information A 2 ATHENA ZEUS LQ HQ LQ/LQ (23, 9) (18, 14) LQ/HQ (23, 9) (20, 12) HQ/LQ (18, 14) HQ/HQ (18, 14) (20, 12) 2 Nash equilibria: Athena: Always choose different than Zeus (LQ, HQ/LQ), (HQ, HQ/LQ) Value for Zeus 18, value for Athena 14

Different Assumptions about information A 3 • Zeus has to make production decision earlier. • But before it conducts a very thorough and costly market research, which will allow to determine whether the market is small or big • Athena will not know the outcome of this research but will know that it took place

Different Assumptions about information A 3 ATHENA LQ/LQ LQ/HQ ZEUS HQ/LQ HQ/HQ LQ/LQ (23, 9) (16, 16) (25, 7) (18, 14) LQ/HQ (23, 9) (20, 12) HQ/LQ (18, 14) (12, 20) (24, 8) (18, 14) HQ/HQ (18, 14) (16, 16) (22, 10) (20, 12) ATHENA LQ/LQ LQ/HQ ZEUS HQ/LQ HQ/HQ LQ/LQ (23, 9) (16, 16) (25, 7) (18, 14) LQ/HQ (23, 9) (20, 12)

Different Assumptions about information A 3 ATHENA LQ/LQ LQ/HQ ZEUS HQ/LQ HQ/HQ LQ/LQ (23, 9) (16, 16) (25, 7) (18, 14) LQ/HQ (23, 9) (20, 12) HQ/LQ (18, 14) (12, 20) (24, 8) (18, 14) HQ/HQ (18, 14) (16, 16) (22, 10) (20, 12) Nash equilibrium: Zeus (dominant strategy): produce HQ systems if market is small; produce LQ systems if market is big. Athena: always produce HQ systems (HQ/LQ, HQ/HQ) Value for Zeus: 22, value for Athena: 10

Different Assumptions about information A 3 • Interesting: Athena’s strategy changed from “always choose different than Zeus” to “always choose HQ systems”, despite the fact that the only new information for her was that Zeus has conducted market research. – The reasoning is as follows: Since Zeus chose HQ, Athena wins from choosing LQ only in case the market is big. Since Athena knows about market research, it is clear for her that since Zeus chose HQ, it cannot be that the market is big – and hence in this situation it is better for Athena to produce HQ • It requires quite a subtle reasoning. • Zeus increased the profits from 18 to 22. So market research is profitable if its cost does not exceed 4.

Different Assumptions about information A 4 • What if Athena also knew the results of the market research – the same as at the beginning

Different Assumptions about information A 5 • What if Zeus hides from Athena that it conducted a market research? • Athena will have wrong idea about the game being played. – Athena will think that the game is as in A 2 – Zeus will know that the right game is as in A 3 • Athena plays HQ/LQ (optimal in A 2) – Zeus will exploit that an play HQ/LQ and will get 24 instead of 22 • Making market research secret brings Zeus another $2 mln.

Duopol Stackelberga (1934) • Tak samo jak w przypadku konkurencji ilościowej 2 firm wg Cournot: – Tylko zamiast jednocześnie, jedna z firm może zdecydować bądź zobowiązać się wcześniej niż inna co do produkcji • Model nazywa się wówczas oligopolem Stackelberga

Model Cournot - przypomnienie • Zysk ze sprzedaży i-tej firmy • Najlepsze odpowiedzi graczy: • Równowaga i zyski w równowadze:

Exercise • Two firms produce identical good. • Each firm decides upon its production levels. • Inverse demand is p(x 1, x 2)=20 -2 x 1 -2 x 2 (or 0 if x 1+x 2>10) • Cost function is the same for both players c(xi)=8 xi, i=1, 2 1. Cournot duopoly – determine Nash equilibrium

Model Stackelberga 1 • Gra ma dwa etapy: – Firma 1 wybiera poziom produkcji (lub zdolności produkcyjnych) q 1 ≥ 0 – Firma 2 widzi wybór pierwszej i również wybiera poziom produkcji q 2 ≥ 0 • Zyski firm są takie same q 1 2 q 2 Π 1, Π 2

Równowaga indukcji wstecznej • Racjonalność sekwencyjna firmy II wymaga, że będzie najlepiej odpowiadać na jakikolwiek wybór q 1: • Firma I natomiast wybierze q 1 tak, aby maksymalizować:

Równowaga indukcji wstecznej • Równowaga indukcji wstecznej • Wynik gry w równowadze • Zyski w równowadze – korzyść lidera First-mover advantage

Exercise continued 2. Player 1 decides first, Player 2 after observing what player 1 has done – The structure of the game is common knowledge a) Find SPNE b) Consider the following pair of strategies: x 1=4, x 2=2 if x 1<4; x 2=1 if x 1=4; x 2=1. 5 if x 1>4. Is this a (imperfect) Nash Equilibrium? c) Show that a pair of strategies: x 1=3; x 2=1. 5 irrespective of an observed x 1 is not a NE.

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