Pragmatics & Game Theory Session 9: The IBR-Model

Transcription

1 Pragmatics & Game Theory Session 9: WiSe 3/4

2 Table of Content Introduction Introduction How to compute the IBR-sequence Homeworks 2 The Generalized M-Implicature The 'Some-But-Not-All' Game 3 The 'Or-Reading' Game The 'Human' Game

3 How to compute the IBR-sequence How to compute the IBR-sequence Homeworks Given: a signaling game and a starting strategy Wanted: the IBR-sequence and the nal x-point strategy pair Approach: apply the following 'Workable Rules' until a circle is reached (a strategy reoccurs) Workable Rules Mirror the strategy 2 Add missing edges (R) Surprise messages: each true state equiprobable (S) Unexpected states: each true message equiprobable 3 Remove weak alternatives (S,R) Remove edges with lower probability values (S) Remove edges that accuse higher costs 4 Update probabilities ((S) multiplied by prior probability)

4 How to compute the IBR-sequence Homeworks How to compute the IBR-sequence: some-all game Workable Rules Mirror the strategy 2 Add missing edges (R) Surprise messages: each true state equiprobable (S) Unexpected states: each true message equiprobable 3 Remove weak alternatives (S,R) Remove edges with lower probability values (S) Remove edges that accuse higher costs 4 Update probabilities ((S) multiplied by prior probability) S 0 R S 2 R 3 t.25 m a a/t.5 m a a/t.25 t.5 m s a/t.5 m s a/t

5 How to compute the IBR-sequence Homeworks How to compute the IBR-sequence: DOPL game Workable Rules Mirror the strategy 2 Add missing edges (R) Surprise messages: each true state equiprobable (S) Unexpected states: each true message equiprobable 3 Remove weak alternatives (S,R) Remove edges with lower probability values (S) Remove edges that accuse higher costs 4 Update probabilities ((S) reconsider prior probability) t f S m u R a/t f S 2.8 m u R 3 a/t f S 4.8 m u R 5 a/t f t r.. m m.2.5 a/t r m m a/t.5 r.2 m m a/t r

6 How to compute the IBR-sequence Homeworks How to compute the IBR-sequence: milk game Workable Rules Mirror the strategy 2 Add missing edges (R) Surprise messages: each true state equiprobable (S) Unexpected states: each true message equiprobable 3 Remove weak alternatives (S,R) Remove edges with lower probability values (S) Remove edges that accuse higher costs 4 Update probabilities ((S) multiplied by prior probability) S 0 R S 2 R 3 t c.4.4 m c m m a/t c.8 m c m m a/t c. t g. m g a/t g m g.2 a/t g

7 Homeworks Exercise & 2 How to compute the IBR-sequence Homeworks Table depicts the 'try-want-succeed' game. Draw the initial sender strategy S 0 of the game. 2 Draw the IBR-sequence starting with S 0. What is the x point strategy? Pr(t) a σ a τ σ a ω τ σ m want m try m suc t σ /3, 0,0 0,0 t τ σ /3 0,0, 0,0 t ω τ σ /3 0,0 0,0, Table: Parameters of the try-want-succeed game S 0 R S 2 t σ /9 /9 /9 t τ σ /6 /6 m suc m try a/tσ a/t τ σ /3 /3 m suc m try t ω τ σ /3 m want a/t ω τ σ /3 m want

8 Homeworks Exercise 3 & 4 How to compute the IBR-sequence Homeworks Table 2 depicts the extended milk game. 3 Draw the initial sender strategy S 0 of the game. 4 Draw the IBR-sequence starting with S 0. What is the x point strategy? Pr a cmk a gmk a ccmk a smk m mk m cmk m gmk m ccmk m smk t cmk.7, 0,0 0,0 0,0 t gmk. 0,0, 0,0 0,0 t ccmk. 0,0 0,0, 0,0 t smk. 0,0 0,0 0,0, Table: Parameters of the extended milk game Result: S0 = t cmk m mk, m cmk t gmk m mk, m gmk t ccmk m mk, m ccmk t smk m mk, m smk R = m mk a cmk m cmk a cmk m gmk a gmk m ccmk a ccmk m smk a smk S 2 = t cmk m mk t gmk m gmk t ccmk m ccmk t smk m smk

9 Homeworks Question 5 & 6 How to compute the IBR-sequence Homeworks Read Signal to Act, Chapter 2 (pp 53-65) and answer: 5 What does the property 'responding to unbiased beliefs' mean? If my belief about my participants behavior has multiple possibilities for a given choice point, I believe each possibility as equiprobable, therefore unbiased. If I play the best response to such a belief, I am 'responding to unbiased beliefs'. 6 What kind games accuse a circle in the IBR-sequence? And what kind of games reach a x point? Since the number of possible strategies of an IBR-sequence is countable for any game, there must be a recurrence of strategies at one point. Thus, each game accuses a circle in the IBR-sequence. A x point is a circle of length, which generally emerge for games with aligned interests.

10 M-Implicature with 3 states The Generalized M-Implicature The 'Some-But-Not-All' Game Pr(t) a f a r a sr m u m m m c t f.6, 0,0 0,0 t r.3 0,0, 0,0 t sr. 0,0 0,0, Table: Paramteters of the game S 0 = t f m u, m m, m c t r m u, m m, m c R = m u a f m m a f t sr m u, m m, m c m c a f

11 M-Implicature with 3 states The Generalized M-Implicature The 'Some-But-Not-All' Game S 0 = S 2 = S 4 = t f m u, m m, m c t r m u, m m, m c R = t sr m u, m m, m c S 6 = t f m u t r m u t sr m u t f m u t r m m t sr m m m u a f m u a f m m a f m c a f R3 = m m a f, a r, a sr t f m u t r m m m c a f, a r, a sr m u a f R5 = m m a r m c a f, a r, a sr R7 = m u a f m m a r t sr m c m c a sr

12 Generalized M-Implicature The Generalized M-Implicature The 'Some-But-Not-All' Game For M-Implicatures with n states/messages/actions, the IBR-model makes the prediction of n separate allocations between information and message Beaver and Lee (2004) argue that the IBR-model is actually wrong, because it overgenerates: in natural language there is no support for such a strong prediction Franke (2009) argues that the IBR-sequence depicts a course of idealized pragmatic reasoning that does not take bounded rationality (of a few steps) into consideration Note that a M-Implicature of degree n needs 2 n + deliberation steps to reach a x point By stopping after around 4-5 steps, we reach a reasonable strategy of the division of pragmatic labor

13 The 'Some-But-Not-All' Game The Generalized M-Implicature The 'Some-But-Not-All' Game Pr(t) a a m all m some m sbna t / 2, 0,0 t / 2 0,0, Table: Paramteters of the game IBR-sequence with naive start [ ] t m S 0 = all, m some t m some, m sbna m all a R = m some a, a m sbna a [ ] t m S 2 = all t m sbna IBR-sequence with 'cheap' start [ ] t m S 0 = all, m some t m some m all a R = m some a m sbna a [ ] t m S 2 = all t m some

14 The 'Some-But-Not-All' Game The Generalized M-Implicature The 'Some-But-Not-All' Game Situations can be modeled in dierent ways: e.g. the scalar implicature can take the more concrete, but longer expression into consideration The IBR-model in its Vanilla version does not make the right prediction (in terms of the phenomenon) One way out would be to start with a cheap sender strategy Another way would be to have a more rational treatment for surprise messages (Franke 2009)

15 The 'Or-Reading' Game The 'Or-Reading' Game The 'Human' Game What is the interpretation of Take an apple or a banana.? Pr(t) a A a B a AB m A m B m AorB m AandB t A / 3, 0,0 0,0 t B / 3 0,0, 0,0 t AB / 3 0,0 0,0, Table: Paramteters of the game The vanilla version of the IBR-model results in the inclusive reading We get the exclusive reading by changing the treatment of surprise messages (previous level) Another approach: lifting the game (Franke 2009)

16 The 'Human' Game Introduction The 'Or-Reading' Game The 'Human' Game Pr a w a m a g a b m h m a m c m w m m m g m b t w.4, 0,0 0,0 0,0 t m.4 0,0, 0,0 0,0 t g. 0,0 0,0, 0,0 t b. 0,0 0,0 0,0, The game depicts a hypernym/hyponym-structure, which does not entail a specic generalized implicature reading The IBR-model also predicts no specic/optimalized reading, but only the semantic reading (for the receiver) The IBR-model predicts a precise strategy for the sender

17 Conclusion Introduction The 'Or-Reading' Game The 'Human' Game The Workable Rules are a set of rules that help detecting the IBR-sequence of the vanilla IBR-model The vanilla IBR-model makes the right prediction for the scalar implicature (of any scale lengths) the I-implicature (for any number of more concrete information states) the division of pragmatic labor (it overgenerates for more 'ne-grained' types of M-implicatures, what can be moderated by reconsidering bounded rationality) The vanilla IBR-model makes the wrong prediction for the 'some-but-not-all' game and the 'or reading' game In both cases the treatment of surprise messages is tipping the scales The vanilla IBR-model makes the right prediction for the 'human' game by generating the semantic reading

18 Introduction The 'Or-Reading' Game The 'Human' Game