EC3224 Autumn Lectue #5 Extensive Fom Games with Pefect Infomation Reading Osbone, Chaptes 5, 6, 7.1., 7.2, 7.7 Leaning outcomes constuct simple games in extensive fom undestand the concept of a subgame-pefect equilibium fo such games find the subgame-pefect equilibium though backwad induction
Timing mattes Conside a game of ock, pape, scissos Simultaneous moves! How do childen often cheat in that game?
Extensive Fom Games The stategic fom of a game does not epesent the timing of moves Hence plans of actions ae fixed and cannot be changed In contast, extensive fom games captue the sequential stuctue of a game This captues that playes can change thei plans Fo now, we conside extensive fom games with pefect infomation, i.e. when choosing an action a playe knows the actions chosen by playes moving befoe he
Histoies and Teminal Histoies A sequence of actions in an extensive fom game fo which no actions follow is called a teminal histoy if (a 1, a 2,, a k ) is a teminal histoy, than any (a 1, a 2,, a m ) with m k is a subhistoy and fo m < k a pope subhistoy (including empty sequence Ø) A histoy is any subhistoy of any teminal histoy
Extensive Fom Games An extensive fom game is chaacteized by playes a set of teminal histoies (that is all possible complete sets of actions in the game) a playe function that assigns to any pope subhistoy of any teminal histoy the playe who moves afte this histoy pefeences ove the set of teminal histoies
Game Tees An extensive fom game can be epesented in a game tee This shows who moves when (at the nodes, epesenting the (non-teminal) histoies) thei available actions (the banches) and the payoffs epesenting pefeences ove teminal histoies (at the teminal nodes)
A Game Tee Playe 1
A Game Tee Playe 1 Left
A Game Tee Playe 1 Left Right
A Game Tee Playe 1 Left Right Playe 2
A Game Tee Playe 1 Left Right Playe 2 l
A Game Tee Playe 1 Left Right Playe 2 l
A Game Tee Playe 1 Left Right Playe 2 Playe 2 l l
A Game Tee The pefeences can be epesented by a payoff function ove the teminal histoies Playe 1 Left Right Playe 2 Playe 2 l l U 1 (L,l) U 2 (L,l)
A Game Tee The pefeences can be epesented by a payoff function ove the teminal histoies Playe 1 Left Right Playe 2 Playe 2 l l U 1 (L,l) U 1 (L,) U 2 (L,l) U 2 (L,)
A Game Tee The pefeences can be epesented by a payoff function ove the teminal histoies Playe 1 Left Right Playe 2 Playe 2 l l U 1 (L,l) U 1 (L,) U 1 (R,l) U 1 (R,) U 2 (L,l) U 2 (L,) U 2 (R,l) U 2 (R,)
Stategies A stategy is a complete desciption of a playe s actions at all the nodes when it s his tun to move, e.g. fo playe 2 to choose afte L and l afte R. Playe 2 has 4 stategies: {(l,l),(l,),(,l),(,)} Mixed stategies ae defined as usual Playe 1 Left Right Playe 2 Playe 2 l l U 1 (L,l) U 1 (L,) U 1 (R,l) U 1 (R,) U 2 (L,l) U 2 (L,) U 2 (R,l) U 2 (R,)
Nash Equilibium The Nash equilibium of an extensive fom game is defined as usual: s* is a (mixed-stategy) Nash equilibium if fo evey playe i and evey mixed stategy s i : U i (s*) U i (s i, s -i *) whee U i (s) is i s expected payoff fo the teminal histoies that ae induced by the playes following stategies s Nash equilibium is not a satisfactoy concept fo extensive fom games: moves afte histoies that would not be eached ae ielevant fo the outcome, so in Nash equilibium, the actions do not have to be payoff maximizing at these points but what if a playe makes a mistake and the histoy is eached?
Example: Mini Ultimatum Game Popose (Playe 1) can suggest one of two splits of 1: (5,5) and (9,1). Responde (Playe 2) can decide whethe to accept o eject (9,1), but has to accept (5,5). Reject leads to fo both Playe 1 (9,1) (5,5) Playe 2 a 9 1 5 5
Mini Ultimatum Game in Stategic Fom Playe 2 accept (9,1) eject (9,1) Playe 1 popose (5,5) 5,5 5,5 popose (9,1) 9,1, Thee ae two equilibia: 1. (popose (9,1), accept (9,1)) 2. (popose (5,5), eject (9,1)). Equilibium 2 is in weakly dominated stategies (eject (9,1) is weakly dominated)
Example: Mini Ultimatum Game In stategic fom, we assume playes have coect beliefs due to expeience But if 1 always poposes (5,5), she does not gain expeience afte popose (9,1) Expeience could esult fom occasional mistakes But if 1 occasionally mistakenly chooses (9,1), then eject (9,1) is not optimal any moe In extensive fom it is seen that equilibium 2 is not convincing because it elies on a non-cedible theat: if the 1 poposes (9,1) playe 2 has an incentive to deviate (i.e. to accept)
Sequential Rationality Think about equilibia in a game in extensive fom What should we do afte a histoy that will not occu in equilibium? Following the definition of Nash-equilibium, the choice is ielevant afte this histoy Howeve, an equilibium appeas to be moe convincing, if we equie that each playe chooses optimally in any decision node and takes into account that all playes will do so in the futue We call this sequential ationality This notion is captued by subgame pefect equilibium (Selten, 1965)
Nobel Pize in Economics, 1994 QuickTime and a decompesso ae needed to see this pictue. QuickTime and a decompesso ae needed to see this pictue. Reinhad Selten (left), John Hasanyi (ight, with the King of Sweden) and
John F. Nash QuickTime and a decompesso ae needed to see this pictue. aka the Gladiato
Subgames A subgame is a pat of an extensive fom game following some histoy, with the playe function and pefeences as fo the whole game A subgame is a game in itself: it stats at a single node it contains all moves of the whole game following the histoy once we ae in a subgame, we do not leave it By convention, we conside the entie game to be a subgame of itself
Subgame Pefect equilibium A stategy s fo a game induces a stategy s(g) fo any subgame g A subgame pefect Nash equilibium (SPNE) is a set of stategies {s i, i=1,,n} such that fo each subgame g, the set of induced stategies {s i (g), i=1,,n} foms a Nash equilibium fo this subgame, that is, in no subgame can a playe incease he payoff by deviating to anothe stategy. In paticula, since a game is a subgame of itself, a SPNE is always a Nash equilibium
Subgame Pefect equilibium In a SPNE the past is ielevant, i.e. howeve we got to the subgame, we have to play an equilibium, even if the stategies imply that we do not each this subgame This is quite a stong equiement, because even if we got to the subgame by clealy non-ational behavio, we still equie ationality fo the futue Futhemoe, in a SPNE, all playes must have the same expectation concening the equilibium to be played in a subgame, which is not necessaily always easonable
Example: Mini Ultimatum Game Thee ae 2 subgames: whole game and following (9,1) If 1 poposes (9,1), 2 is bette off accepting Given that, 1 is bette off poposing (9,1) Thus the only subgame pefect equilibium is: (popose (9,1), accept (9,1)) Playe 1 (9,1) (5,5) Playe 2 a 9 1 5 5
Backwad Induction Length of a subgame: numbe of moves in longest teminal histoy A game has a finite hoizon if the length of the longest teminal histoy is finite Subgame pefect equilibia of games with finite hoizon can be found by backwad induction Backwad induction: fist find the optimal actions in the subgames of length 1 then taking these actions as given, find the optimal actions of playes who move at beginning of subgames of length 2 continue woking backwads, ends afte finitely many steps If in each subgame thee is only one optimal action, this pocedue leads to a unique subgame pefect equilibium
Example: Mini Ultimatum Game Thee is one subgame of length 1, following (9,1) Playe 1 (9,1) (5,5) Playe 2 a 9 1 5 5
Example: Mini Ultimatum Game Thee is one subgame of length 1, following (9,1) The optimal action is accept Playe 1 (9,1) (5,5) Playe 2 a 9 1 5 5
Example: Mini Ultimatum Game Thee is one subgame of length 1, following (9,1) The optimal action is accept Thee is one subgame of length 2, the whole game Playe 1 (9,1) (5,5) Playe 2 a 9 1 5 5
Example: Mini Ultimatum Game Thee is one subgame of length 1, following (9,1) The optimal action is accept Thee is one subgame of length 2, the whole game Taking accept in the subgame of length 1 as given, we see that (9,1) is optimal Playe 1 (9,1) (5,5) Playe 2 a 9 1 5 5
The Ultimatum Game Popose (Playe 1) suggest (intege) split of a fixed pie, say 1. Responde (Playe 2) accepts (poposal is implemented) o ejects (both eceive ) Thee is no unique solution fo the subgame following (1,) Playe 1 Playe 2 (1,) (9,1) (5,5) (,1) a a a a 1 9 1 5 5 1
Genealizing backwad induction If at one step, thee is no unique best action of a playe, backwad induction as above does not wok We can genealize: at each step, conside the set of optimal actions then follow each of them as above This pocedue allows us to find all subgame pefect equilibia of a game with finite hoizon Apply this to ultimatum game: Fo subgames of length 1, fo each offe (1 - x, x) with x >, the esponde stictly pefes to accept fo (1,) esponde is indiffeent, so thee ae two optimal actions, accept and eject So thee ae two subgame pefect equilibia: 1. The popose offes (1,) and the esponde accepts all offes, including (1,) 2. The popose offes (9,1) and the esponde ejects (1,) but accepts all positive offes, i.e. accepts (9,1), (8,2) etc
Existence of subgame pefect equilibium A game is finite if it has a finite hoizon and finitely many teminal histoies (i.e. each time a playe moves, he has finitely many available actions) If a playe has finitely many actions, we can find (at least) one that is optimal Thus thee is always a solution at each step of backwad induction and thus Poposition: Evey finite extensive game with pefect infomation has a subgame pefect equilibium
Futhe genealizing backwad induction Assume afte playe 1 chooses R, both playes move once moe, but simultaneously (1 chooses ow, 2 column) Thus we have a subgame afte R and subgame pefect equilibium equies equilibium play in this subgame By stating at the end, we find SPNE ((A,U),(a,L)) and ((B,D),(a,R) Playe 1 A B Playe 2 L R a b U 2,2 3,1 3 1 D 1,3 4,4
Backwad Induction and Iteated Elimination of Dominated Stategies Backwad Induction coesponds to iteated elimination of dominated stategies in the stategic fom of the game: a stategy that is not payoff maximizing in a subgame of length 1 is dominated by the stategy that yields a highe payoff in this subgame but is othewise identical picking the payoff-maximizing choice in this subgame coesponds to eliminating the dominated stategies having eliminated non-maximizing stategies in subgames of length 1, a stategy that is not payoff maximizing in a subgame of length 2 is dominated by one that is continuing in this fashion leads exactly to the same solution as backwad induction
Commitment In some situations a playe can pofit fom educing his options Hee, Playe 2 would like to eliminate option l afte Right This would lead Playe 1 to choose Left Hence Playe 2 would pofit fom being able to commit to choosing afte Right But the ability to commit would be pat of the game Playe 1 Left Right Playe 2 Playe 2 l l 5 8 5 2
Commitment In some situations a playe can pofit fom educing his options Hee, Playe 2 would like to eliminate option l afte Right This would lead Playe 1 to choose Left Hence Playe 2 would pofit fom being able to commit to choosing afte Right But the ability to commit would be pat of the game Playe 1 Left Right Playe 2 Playe 2 l 5 5
Poblems with Backwad Induction If thee ae many playes, the demands on the playes ationality become vey stong. Example: 1 playes 1 P 2 P... 1 P 1 1 T T T 1 99 1 99 99
Poblems with Backwad Induction If a playe moves epeatedly and has aleady violated the backwad induction solution once, what shall the othes think? Example: centipede game 1 P 2 P 1 P 2 P 1 P 2 P 256 T T T T T T 64 4 2 16 8 64 32 1 8 4 32 16 128
Application: Agenda Contol and Stategic Voting 3 playes ae voting upon thee poposals, x,y,z Pefeences: P1: u(x) > u(y) > u(z) P2: u(y) > u(z) > u(x) P3: u(z) > u(x) > u(y) It is ageed that they will vote in the following way: fist two poposals ae voted upon then the winning poposal and the emaining poposal ae voted upon Assume P1 can detemine the sequence of votes (the agenda) then if all votes ae tuthful, she will choose fist y against z and then the winne (y) against x, then x wins But tuthful voting is not SPNE: in the last stage, voting will be tuthful, but knowing that, P2 should vote fo z in the fist stage (P2 will vote stategically) in the SPNE, P1 will choose fo the fist ound x and z, then P3 will vote stategically fo x (because othewise y will win in the end) and then x will win against y in the second stage So by being able to detemine the agenda, P1 can get pefeed esult
Poblem set #5 1. Conside the centipede game a) Find the subgame pefect equilibium though backwad induction b) What is the poblem fo playe 2 if playe 1 chooses I on his fist move? c) How would you play the game? 2. Do question 1 (a) and (c) fo the 1-playes game. 3. Osbone 163.2 4. Osbone 173.3 5. (Osbone 173.4) 6. (Osbone 176.1) 7. (Osbone 177.1) ESSAY TOPIC (max 15 wods): Choose a ecent stoy fom the news. Descibe the situation as a game (in stategic o extensive fom). Find the equilibium. Discuss whethe actual events coespond to the equilibium (if not discuss possible easons why not). Discuss you assumptions. (if you e eally out of ideas, ty A Beautiful Mind, but watch out, the descibed eq. is wong)