Game Theory forward induction and application

1. Game Theory forward induction and application Univ. Prof.dr. M.C.W. Janssen University of Vienna Winter semester 2011-12 Week 45 (November 7)

2. Battle of the Sexes with Outside option • What are the SPE in pure strategies? • (PlayB,B) • (OutF,F) • Are both equilibria reasonable? • Is there something player 2 should conclude from the fact that player 1 decided to play the BS game? • Indication of likely future play? 2, 2 out 1 play

3. Forward Induction - Idea • Player 1 could get a safe pay-off of 2 • There is only one way he can get a higher pay-off when he chooses to play BS • Namely when outcome is B,B • Thus, when player 1 decides to play BS he intends to play B in the BS game • Player 2 can figure this out and expects player 1 to play B in BS game and thinks what his best response to B would be (which is B) • Knowing this, player 1 can safely play BS game and choose B • Conclusion: Past actions may sometimes signal how player intends to play in future

4. Normal form representation • IEWDS: • Play F is strongly dominated by Out • Given this, F is weakly dominated by B • Given this, Out is strongly dominated by PlayB • Solution is the Forward Induction outcome

5. Application • Consider the following two-stage game • 1st stage: auction where N players can choose to bid in cents; first-price sealed-bid auction • 2nd stage: market game played by 2 winners with a low and a high price equilibrium • What is a strategy? • What are SPE? • What is the equilibrium consistent with forward induction?

6. SPE in application • Many SPE. In particular, both high and low price can be supported in a SPE: • Bid a number x between 7 and 10 and play H if (x,x) is played; otherwise play L • Bid the largest number in cents below 6 and play L afterwards • Check that these are SPE

7. Unique equilibrium consistent with forward induction I • Bid the largest number in cents below 10 and play H afterwards • IEDS rules out any bid below 5.99 • If you bid this number pay-off is either 0 (if two bidders bid larger), or at most 4.01*(2/N) and never negative • Now consider you bid more than 7. • Bidding more than 7 and playing L is dominated • Thus (crucial point), if the other winning player has bid more than 7, he is expected to play H and therefore it is optimal to also play H • To bid slightly more than 7 and play H therefore gives a

8. Unique equilibrium consistent with forward induction II • To bid slightly more than 7 and play H therefore gives a pay-off of • 0 if at least two other bidders bid more (in this case bidding less will result in same pay-off) • Almost 3/2 if one bidder bids the same amount and others bid lower (in this case bidding almost 6 will result in a pay-off of at most 4/(N-1)) • Positive if at least two other bidders bid the same number (in this case bidding less will result in pay-off of 0) • Almost 3 if all other bidders bid lower (in this case bidding almost 6 will result in a pay-off of at most 4*(2/N) • If N is large enough, strategies with “bidding almost 6” are (weakly) dominated • Once, playing L can’t be part of equilibrium (as bids are too high), one can continue with IEDS.

9. Auction Conclusion • Auctions may lead to price collusion in aftermarket • auction forces bidders to bid high • Bidding even higher can only be rationalized by a collusive equilibrium in aftermarket • After such a bid (interpreted as a signal of future intentions), it is optimal to collude for competitors as well (if this is part of equilibrium behavior) • If it is interpreted in this way, it is rational to give this signal.