Somebody’s got to do it

Problem 8.19

Subgame perfect strategy profiles Val: Stall/Stall/ Do it Earl: Do it/ Do it Course of play: Val stalls, Earl does it. Game ends.

Are there other Nash Equilibria?

Some others Suppose Earl’s strategy is Stall/Stall. What is Val’s best response? If Val does this, what are Earl’s best responses? What if Val’s strategy is Stall/Stall/Stall?

Problem 8.16 Nick and Rachel divide 4 candy bars. They take turns choosing. Nick goes first. What should Nick choose first? Preferences are: For Nick For Rachel Snickers Milky Way Milky Way Kit Kat Kit Kat Baby Ruth Baby Ruth Snickers Hint: No matter what happens, Nick will get two bars. Rachel will never choose Snickers.

Nick BR MW KK Sn Rachel Rachel Rachel Rachel MW BR Nick KK Sn BR KK Nick Nick Nick Nick Nick BR KK MW MW BR KK Sn KK BR KK Sn BR 6 11 8 10 9 6 11 9 8 10 9 7 6 9 5 7 7 6 6

Nick BR MW KK Sn Rachel Rachel Rachel Rachel MW BR Nick KK Sn BR KK Nick Nick Nick Nick Nick BR KK MW MW BR KK Sn KK BR KK Sn BR 6 11 8 10 9 6 11 9 8 10 9 7 6 9 5 7 7 6 6

Nick BR MW KK Sn Rachel Rachel Rachel Rachel MW BR Nick KK Sn BR KK Nick Nick Nick Nick Nick BR KK MW MW BR KK Sn KK BR KK Sn BR 6 11 8 10 9 6 11 9 8 10 9 7 6 9 5 7 7 6 6 In the simplified game, where Nick’s first move must be Sn or MW, Subgame perfect equilibria: Nick plays MW/KK/MW/MW/Sn/Sn/KK Rachel plays MW/BR or MW/KK

Return to Alice-Bob story Go to Movies Bob Go shoot pool Alice Go to A Go to B Alice Alice Go to A Go to B Go to B Go to A Go to A Go to B 2 3 2.5 1 2.5 1 3 2 Alice knows whether Bob is shooting pool before she decides which movie to go to.

Solving for SPNE Identify the proper subgames. Find their SPNE’s SPNE for full game is a strategy profile whose substrategy profiles are SPNE

In our game. Two proper subgames. One starts where Alice discovers that Bob is shooting pool. One starts where Alice discovers that Bob is going to the movies. These have no proper subgames, so any NE for these subgames is subgame perfect So our next step is to find Nash equilibria for each.

Return to Alice-Bob story Go to Movies Bob Go shoot pool Alice Go to A Go to B Alice Alice Go to A Go to B Go to B Go to A Go to A Go to B 2 3 2.5 1 2.5 1 3 2 Alice knows whether Bob is shooting pool before she decides which movie to go to.

One subgame In the subgame where Alice decides what to do when Bob shoots pool, the NE is for Alice to go to Movie A.

The other subgame in strategic form Alice Movie A Movie B 2,3 0,0 1,1 3,2 Bob Pure strategy equilibria where both go to A and where both go to B. Mixed strategy equilibrium where Alice goes to movie A with probability q Such that Bob is indifferent between the two movies. That is, 2q+0(1-q)=1q+3(1-q), which implies q=2/3. Similar calculation shows that Alice is indifferent between the two movies when Bob goes to Movie B with probability 2/3. Expected payoff to Bob in mixed strategy N.E. Is 2q=4/3.

Subgame perfect NE strategy profiles For Bob, Movies, Movie B if movies. For Alice, A if Bob shoots pool, B if Bob goes to movies For Bob, Shoot pool, Movie A if movies. For Alice, A if Bob shoots pool, A if Bob goes to movies. For Bob, Shoot pool, Randomize with probability 2/3 of going to B if movies. For Alice, Movie A if Bob shoots pool, Randomize with probability 2/3 of going to A if movies

Wait a minute… Suppose Bob chooses to go to the movies rather than to shoot pool. What can Alice conclude that he expects her to do? If he expects this what should she do? So doesn’t that leave us with only one plausible equilibrium? Notion of forward induction…

Introduction to Induction Backward induction. Draw conclusions about what other player(s) will do by working back from the end and assuming they are rational. Forward induction. Drawing conclusions about what other player(s) believe by looking at what they have done and assuming they are rational.

Problem 7.13 How many pure strategy Nash equilibria does this game have? Player 2 Player 1 Left Right Top 1, 2 0,2 Bottom 1,0 3,4

Mixed strategies? How many Nash equilibria does it have, including all mixed strategies? Player 2 Player 1 A) 0 B) 1 C) 2 D) 3 E) 4 Left Right Top 1, 2 0,2 Bottom 1,0 3,4

Only the Pure This game has no mixed Nash equilibria that are not pure Player 2 Player 1 Note: Bottom weakly dominates top for Player 1. Right weakly dominates Left for Player 2 Player 1 will mix only if Player 2 plays Left for sure. But if Player 1 mixes, Player 2’s best response is Right. So we can’t have Player 1 mixing. Player 2 will mix only if Player 1 plays Top for sure. But if Player 2 mixes, Player 1’s best response is Bottom. So we can’t have Player 2 mixing. Left Right Top 1, 2 0,2 Bottom 1,0 3,4

How does number of pure and mixed strategy equilibria vary with X? Player 2 Player 1 How many Nash equilibria are there if X>2? A) 0 B) 1 C) 2 D)3 E) 4 Left Right Top 1, 2 0,X Bottom 1,0 3,4

Suppose X<2? There are 2 pure strategy N.E. Player 2 Player 1 There are 2 pure strategy N.E. Bottom weakly dominates top for Player 1, but there is no weak domination for Player 2. Player 1 will mix only if she is sure that Player 2 will go left. Left Right Top 1, 2 0,X Bottom 1,0 3,4

Suppose X<2? Player 2 Player 1 Player 1 will mix only if she is sure that Player 2 will go left. Let p be the probability Player 1 goes Top Player 2 will go left only if 2p≥pX+4(1-p) Equivalently (6-X)p≥4 or p≥4/(6-X) Left Right Top 1, 2 0,X Bottom 1,0 3,4

Continued… Player 2 will go left only if 2p≥pX+4(1-p) Equivalently (6-X)p≥4 or p≥4/(6-X) When X<2, there are mixed strategy Nash equilibrium in which Player 1 plays Top with any probability p≥4/(6-x) and Player 2 Plays Left for sure.

Full story For X<2, there are many mixed strategy Nash equilibria and 2 pure strategy Nash equilibria. For X=2, there are 2 pure strategy Nash equilibria and no mixed strategy equilibria that are not pure. For X>2 there is exactly one Nash equilibrium, a pure strategy.

Todd and Steven Divide the Estate Problem 8.10

Bargaining over 100 pounds of gold Round 1: Todd makes offer of Division. Steven accepts or rejects. Round 2: If Steven rejects, estate is reduced to 100d pounds. Steven makes a new offer and Todd accepts or rejects. Round 3: If Todd rejects, estate is reduced to 100d2 pounds. Todd makes new offer and Steven accepts or rejects. If Steven rejects, both get zero.

Working backwards for SPNE In last subgame, Steven must either accept or reject Todd’s offer. If he rejects, both get 0. If he accepts, he gets what Todd offered him. If Todd offers any small positive amount ε, Steven’s best reply is to accept. So in next to last subgame, Todd would offer Steven ε and take 100d2-ε for himself.

Part of game tree Todd Propose Steven Accept Reject Steven Propose

Back one more step At node where Steven has offered Todd a division, there are 100d units to divide. Todd would accept 100d2 or more, would reject less. So at previous node Steven would offer Todd 100d2 and would have 100(d-d2) for himself.

Back once more Now consider the subgame where Todd makes his first proposal. At this point there are 100 pounds of gold to divide. Todd sees that Steven would accept anything greater than 100(d-d2). So Todd would offer Steven 100(d-d2)+ε and keep 100(1-d+d2)-ε for himself.

SPNE Equilibrium strategy for Todd First node: Offer Steven 100(d-d2)+ε Second node: If Steven rejects Todd’s offer and makes a counteroffer to Todd: Accept 100d2 or more, reject less. Third node: If Todd rejects Steven’s counter offer, make a new offer to Steven of a small ε.

SPNE strategy for Steven First node: Accept any offer greater than 100(d-d2), reject smaller offers. Second node: If Steven rejects Todd’s first offer, then offer Todd 100d2 Third node: Accept any positive offer.

Payoffs Suppose d =.9, then Todd offers Steven 100(d-d2)=100(.9-.81)= 9 and keeps 91 for himself. If d=.5, Todd offers Steven 100(d-d2)= 25 and keeps 100-25=75 for himself. Notice that Steven’s share 100(d-d2) is largest when the decay rate is d=.5. What happens with more rounds of bargaining?

Don’t Forget The Purloined Letter