SF2972 Game Theor Problem set on etensive form games Mark Voorneveld There are five eercises, to be handed in at the final lecture (March 0). For a bonus point, please answer all questions; at least half of our answers must be correct. Do not hesitate to contact me with questions. If ou want to draw game trees, I recommend using the ps-tree commands in the PSTricks package for L A TEX. Good luck! It was fun teaching ou! Eercise. (a) Make eercise 03.2 from Osborne and Rubinstein. (b) Find the subgame perfect equilibria if there is an upper bound M N on the numbers that the plaers can announce in the second round. Eercise 2. (a) Find all (pure) Nash and subgame perfect equilibria of the game below. U 2 D U 2 2 U 4 D 4 (4, 6) (7, 5) (8, 7) D (6, 5) U 3 2 D 3 (9, 5) (b) Determine the corresponding strategic game. For each subgame perfect equilibrium, specif an order of iterated elimination of weakl dominated strategies that preserves the corresponding equilibrium. Can the same be done for the game s Nash equilibria?
Eercise 3. Consider the two games below. Out In Out 2 (4, 0) (4, 0) 2 (5, ) (3, 4) (5, ) (3, 4) (a) Do the games have the same (i) strategic form, (ii) reduced strategic form, (iii) agent strategic form? (b) For both games, compute the set of sequential equilibria. (c) Discuss the difference between the two games. What equilibrium (if an) do ou find most appealing?
Eercise 4. In this eercise, we compare consistenc with two other requirements on beliefs which I discussed informall during lecture. A belief sstem µ in an etensive form game with perfect recall is structurall consistent if for each information set I there is a strateg profile β such that: I is reached with positive probabilit under β, µ(i) is derived from β using Baes rule. (a) In Osborne and Rubinstein, Figure 228., specif the set of structurall consistent belief sstems. (b) One could define a new equilibrium notion: an assessment (β, µ) that is sequentiall rational and structurall consistent. Would that make sense? An assessment (β, µ) is weakl consistent if for each information set I that is reached with positive probabilit under β, the beliefs µ(i) over its histories are derived from β using Baes rule. Notice the difference with consistenc: weak consistenc imposes no constraints on beliefs over information sets that are reached with probabilit zero. (c) Consider the information set I of plaer 2 in the game below. The are meant to indicate that whatever else happens in the game tree is irrelevant. c 2/3 /3 2 Which beliefs µ(i) are feasible under (i) consistenc, (ii) structural consistenc, (iii) weak consistenc. Do ou find all beliefs under weak consistenc reasonable? See Osborne and Rubinstein, def. 228.
Eercise 5. I skipped Section 2.3: it introduces unnecessar notation for special cases of etensive form games. This eercise goes through some of the essential parts. (, 3) u u (2, ) L R d d (4, 0) /2 A 2 c 2 (2, 4) u /2 B u (, 0) L R d d (0, ) (, 2) A signalling game is plaed as follows (see the game above for a special case): Chance starts b assigning to plaer, the sender, a tpe. In our eample, the set of tpes is {A, B} and each tpe has equal probabilit. Plaer observes his tpe and sends a message to plaer 2, the receiver. In our eample, the set of messages is {L, R}. Plaer 2 observes the message of plaer (but not s tpe) and chooses an action. In our eample, the set of actions is {u, d}. The game ends. Paoffs can depend on tpes, messages, and actions. This models a range of economic situations in which one plaer controls the information, but another plaer controls the actions. Sometimes the sender might benefit from tring to inform the receiver about his tpe, sometimes it is more beneficial to tr to mislead the receiver. Rather than searching for all sequential equilibria of a signalling game, it is common to make a number of simplifications: firstl, it is common to look at weakl perfect Baesian equilibria in pure strategies (or, more precisel, behavioral strateg profiles where each plaer chooses in each information set a feasible action with probabilit one). Formall, an assessment (β, µ) is a weakl perfect Baesian equilibrium if it is sequentiall rational and weakl consistent. Secondl, the sender ma or ma not be interested in sending information about his tpe: a weakl perfect Baesian equilibrium in pure strategies is called a pooling equilibrium if he assigns to each tpe the same message and a separating equilibrium if he assigns to each tpe a different message. For instance, in a pooling equilibrium of our eample, β (A) = β (B), whereas in a separating equilibrium, β (A) β (B). (a) Find all separating and pooling equilibria of the game above. (b) Which of these equilibria are sequential equilibria?
Eercise a. Game: The game N, H, P, ( i ) i N has N = {, 2}, H = {, Stop, Continue, {(Continue, (, )) :, Z + }}, where (Continue, (, )) denotes the histor where plaer continues and plaers and 2 announce nonnegative integers and, respectivel. P( ) =, P(Continue) = {, 2}, and preferences i over terminal histories represented b the utilit function u i (Stop) =, u i (Continue, (, )) =. Subgame perfect equilibrium: Consider the subgame after Continue. There is no best response to a positive integer; ever nonnegative integer is a best response to 0. It follows that the unique equilibrium of the subgame is with associated paoffs. Anticipating this, plaer chooses Stop to reach paoff vector (, ). Conclude: the subgame perfect equilibrium has plaer plaing (Stop, 0) and plaer 2 plaing 0. Eercise b. Consider the subgame after Continue. The unique best response to a positive integer is M; ever nonnegative integer is a best response to 0. It follows that the subgame has two equilibria: with paoff vector and (M, M) with paoff vector (M 2, M 2 ). If plaer anticipates equilibrium, he prefers to Stop, if he anticipates equilibrium (M, M), he prefers to Continue. So in addition to the subgame perfect equilibrium in a, there is a new subgame perfect equilibrium where plaer plas (Continue, M) and plaer 2 plas M. Eercise 2. The strategic game and its equilibria: The corresponding strategic game is U 2 U 3 U 2 D 3 D 2 U 3 D 2 D 3 U U 4 7, 5 7, 5 4, 6 4, 6 U D 4 8, 7 8, 7 4, 6 4, 6 D U 4 6, 5 9, 5 6, 5 9, 5 D D 4 6, 5 9, 5 6, 5 9, 5 There are seven pure Nash equilibria, marked with (one or two) stars. The subgame perfect equilibria are marked with two stars. Iterated elimination and subgame perfect equilibria: Consider the subgame perfect equilibrium (U D 4, U 2 U 3 ). In the subgame after histor (U, U 2 ), plaer prefers D 4 over U 4 : U U 4 can be eliminated, as it is dominated b U D 4. Knowing that plaer prefers D 4 over U 4, in the subgame after histor U, now prefers U 2 over D 2. So now U 2 U 3 weakl dominates D 2 U 3 and U 2 D 3 weakl dominates D 2 D 3. In the remaining game (where pl. has 3 and pl. 2 has 2 strategies left), no more actions are weakl dominated, so the elimination process stops. Notice that both subgame perfect equilibria have survived the process. Iterated elimination and Nash equilibria: Nash equilibria in which plaer 2 chooses D 2 are alwas eliminated. Eercise 3a(i). No: in the left game, plaer has 4 pure strategies, in the right game onl 3.
Eercise 3a(ii). Yes (up to an arbitrar relabeling of the strategies): in the left game, the pure strategies (Out, ) and (Out, ) of plaer are paoff equivalent and can be replaced b a single one, Out, as in the game on the right. Eercise 3a(iii). No: in the left game, there are 3 information sets, hence 3 plaers in the agent strategic form, in the right game there are 2 information sets, hence 2 plaers in the agent strategic form. Eercise 3b. Let us start with the game on the right (Uhm, wh? Because it s better practice for the eam). The corresponding strategic game is Out 4, 0 4, 0 5, 0, 0 0, 0 3, 4 Since plaer s strateg is strictl dominated b Out, sequential rationalit requires that it is plaed with zero probabilit in a sequential equilibrium: β ( )() = 0. () Distinguish two cases: Case : sequential equilibria with β ( )() > 0: Consistenc requires that in information set {, }, which is reached with positive probabilit, the beliefs are derived from β ( ) using Baes rule. Together with (), this gives that µ({, })() = : plaer 2 believes to be in node with probabilit. Plaer 2 s unique best response to this belief is to choose : β 2 ({, })() =. Consequentl, plaer prefers, giving paoff 5, to Out, giving paoff 4: β ( )() =. We found a candidate sequential equilibrium: (β, β 2, µ) = ((0,, 0), (, 0), (, 0)). To verif consistenc, notice that (β, β 2, µ) is the limit of the sequence of assessments ( ( )) 2/n (/n, 2/n, /n), ( /n, /n), /n, /n. /n Case 2: sequential equilibria with β ( )() = 0: Together with (), this implies that β ( )(Out) =. For notational convenience, denote plaer 2 s strateg β 2 b (p, p) and the belief µ over {, } b (q, q). Sequential rationalit requires that Out is a best response of plaer. He won t pla (see ()) and gives epected paoff 5p, so Out is a best response as long as 5p 4, i.e., as long as p [0, 4/5]. But is ever such p sequentiall rational? If 0 < p 4/5, plaer 2 chooses both actions and with positive probabilit. Both and therefore have to best responses to 2 s beliefs in his information set. This is true onl if q = 4( q), i.e., if q = 4/5. If p = 0, onl action is chosen with positive probabilit. For this to be a best response to plaer 2 s beliefs in his information set, we need that q 4( q), i.e., that q [0, 4/5].
We found the following candidates for sequential equilibria: {(β, β 2, µ) = ((, 0, 0), (p, p), (4/5, /5)) 0 < p 4/5}, and {(β, β 2, µ) = ((, 0, 0), (0, ), (q, q)) 0 q 4/5}. It remains to verif that these assessments are consistent. For the first class of equilibria, let 0 < p 4/5. The assessment is a limit of (β n, β2 n, µ n ) = (( /n, 4/5n, /5n), (p, p), (4/5, /5)). For the second class of equilibria, let 0 q 4/5. The assessment is a limit of 0 + ε 0 q + ε,(ε, ε), (q + ε, q ε), 0 < ε 0. q ε For the game on the left, a similar analsis applies, but it s a bit simpler: conditional on reaching information set {In}, sequential rationalit requires the plaers to choose a Nash equilibrium of the 2 2 game 5, 0, 0 0, 0 3, 4 This game has three Nash equilibria (in the obvious notation): ((, 0), (, 0)) (both choose ), ((0, ), (0, )) (both choose ), and ((4/5, /5), (3/8, 5/8)). Comparing the corresponding equilibrium paoffs, plaer can choose between In and Out. We find three sequential equilibria:. Plaer chooses In with probabilit, with probabilit, plaer 2 chooses with probabilit, and the belief over {(In, ), (In, )} assigns prob. to (In, ). 2. Plaer chooses Out with probabilit, with probabilit, plaer 2 chooses with probabilit, and the belief over {(In, ), (In, )} assigns prob. to (In, ). 3. Plaer chooses Out with probabilit, miture (4/5, /5) over (, ), plaer 2 chooses miture (3/8, 5/8) over (, ), and the belief over {(In, ), (In, )} assigns prob. (4/5, /5) over histories ((In, ), (In, )). Eercise 3c. The games have the same reduced strategic form: if plaer chooses In, he knows that he immediatel has to choose between and. In the second game, this two-step choice is modelled as a single choice. For sequential equilibria, this matters: recall that the are trembling-hand perfect equilibria of the agent strategic form, allowing for mistakes in each information set. This makes it harder to sustain equilibria in the game on the left: there, plaer has two information sets and consequentl two opportunities for mistakes. Which equilibrium ou like most, is a matter of taste. I like the one ending in paoffs (5, ) because of a forward-induction argument. If plaer consciousl gives up secure paoff 4 b not choosing Out, that
must be because he s aiming to get the higher paoff 5 b choosing. Then it makes sense for plaer 2 to choose as well. Eercise 4a. Still to do. Eercise 4b. No, that onl requires plaers to choose rational responses to structurall consistent beliefs. Those beliefs require onl that there is some behavioral strateg from which it can be derived using Baes rule. That behavioral strateg profile need not be related to what plaers actuall do. Eercise 4c. If I is reached with positive probabilit under behavioral strategies β, it follows that plaer chose with positive probabilit, sa p > 0. So the upper node is reached with probabilit 2 3p and the lower node with probabilit 3p. B Baes rule, both consistent and structurall consistent beliefs must assign probabilit 2/3 to the upper and /3 to the lower node. Weak consistenc is more liberal : if plaer chooses with probabilit, there are no constraints on the beliefs over plaer 2 s information set. This seems unrealistic: conditional upon being in that information set, must have chosen (even if that was onl b mistake or as a thought eperiment), so onl the beliefs (2/3, /3) make sense. Eercise 5. Still to do.