Conversational Game Theory Thomas K Harris Graduate Seminar on Dialog Processing November 25, 2003

Conversational Game Theory2 Why look to Game Theory? –“…studying the nature of the rules of games must be useful for the study of grammatical rules, since it is beyond doubt that there is some sort of similarity between them” –L. Wittgenstein (1958) Game Theory Intro –von Neumann, Morgenstein, Nash… A Conversational Game Theory –Power, Houghton, Kowtko and Isard… Conversational Game Theory SDS in Practice SRI Cambridge & later with the EU’s TRINDI Some Evaluation

Conversational Game Theory3 Why Look at Game Theory? Wittgenstein –“The use of a word in the language is its meaning. The grammar describes the use of the words in the language. So it has somewhat the same relation to the language as the description of a game, the rules of the game, have to the game.” Dialogue grammars and user preferences can be coded as game rules and payoffs. Game Theory provides a mechanism and a justification for choosing/predicting among possible utterances (dialogue moves) in a game.

Conversational Game Theory Game Theory Intro

Conversational Game Theory5 Game Theory Origins “Studies the behavior of rational agents in competitive and collaborative situations.” Christos Papadimitriou Conceptualized and clearly defined by John von Neumann c and Little interest until the publication of von Neumann and Morgenstern’s Theory of Games and Economic Behavior [1944].

Conversational Game Theory6 Would you like to play Thermonuclear War? c. 1950’s Military think tanks esp. the Rand Institute become very interested in game theory for logistics, submarine search, air defense… The MAD concept is formalized in game theory. Equilibrium -> Truce A beautiful mind expands the theory from competitive to collaborative games.

Conversational Game Theory7 Are You a Rational Agent? “Studies the behavior of rational agents in competitive and collaborative situations.” The following 6 slides describe an axiomatic treatment of utility for a rational agent. BTW, There’s a related course on this here (Philosophy Dept) : “Rational Choice”

Conversational Game Theory8 Can you consistently order your alternatives? A preference ordering  exists between any two outcomes, and it is transitive.    

Conversational Game Theory9 Are you indifferent to compound lotteries? Compound lotteries can be reduced to simple lotteries. ½ + ½ ( ½ + ½ ) ≡ ½ + ¼ + ¼

Conversational Game Theory10 Are Your Preferences Continuous? Each outcome A i is indifferent to some lottery ticket involving just A 1 and A r, where for each A i, A 1  A i and A i  A r. i.e. There exist a probability p such that p + (1-p) ≡ Note that this says nothing about the value of p other than p  [0,1]. In particular, note that [.5 $ $0 ≡ $2] may be possible (risk aversion, or non-linear value of money). Think for a sec, however, about these three outcomes: $1, ¢1, burning at the stake. What’s your p?

Conversational Game Theory11 Are you indifferent to prize substitutions? If you’ve already claimed an indifference between say, the car and the cash prize, then you should also be indifferent the substitution of one for the other inside a lottery.

Conversational Game Theory12 Are you consistent with lotteries as well as your prizes? Transitivity among lottery tickets applies, that is, If (p1,p2 )  (q1,q2 ) And (q1,q2 )  (r1, r2 ) Then (p1,p2 )  (r1, r2 )

Conversational Game Theory13 Is more of a good thing always better? Lotteries are monotonic. Assuming  (p1, (1-p1) )  (p2, (1-p2) ) if and only if p1 > p2

Conversational Game Theory14 So What? If you answered yes to the last 6 questions, you are a rational agent. This is a minimum set of assumptions for mathematically tractable theories of behavior.

Conversational Game Theory15 Irrational Agents What about fewer assumptions? Mathematical intractability; unsolvable solutions; ambiguous results. Still can be good science, more apt to be called psychology.

Conversational Game Theory16 Super-Rational Agents What about more assumptions? Probably incorrect description of human behavior; overgeneralization of human preferences; sub optimal decisions made on behalf of humans. May still work for games with highly proscribed objectives, e.g. parlor games, or potentially super-rational agents, e.g. virus’s or other simple automata.

Conversational Game Theory17 Ontology of games “Studies the behavior of rational agents in competitive and collaborative situations.” # of players: 2-person, n-person utility relationship: zero-sum, non- cooperative, cooperative information: perfect information, risk, uncertainty

Conversational Game Theory18 Games Chess: 2-player, perfect information, zero- sum Bridge: 2-player!, risk, zero-sum Rock-Paper-Scissors: 2-player, perfect- information, zero-sumRock-Paper-Scissors Prisoners dilemma: 2-player, perfect- information, non-zero-sum

Conversational Game Theory19 A Common Game Tree

Conversational Game Theory Conversational Games

21 Game Theory and Conversation Dialog management is decision making based on utility under uncertainty. This is exactly the domain of Game Theory. Presupposes linguistic “rules” that define how to achieve non-linguistic goals in the context of other players.

Conversational Game Theory22 Conversational Game Types Question Game QW QW-R interrupt pardon RW confirmation

Conversational Game Theory23 Conversational Game Types Pardon Game UnrecognizedPardon

Conversational Game Theory24 Conversational Game Types Confirmation Game Explicit Confirmation Yes Implicit Confirmation Mod No

Conversational Game Theory25 Conversational Game Types Interruption Game Unimportantinformation

Conversational Game Theory26 Conversational Game Types Information Game Information confirmation

Conversational Game Theory27 Conversational Game Types Hello Game Hello

Conversational Game Theory28 An Illustrative Example 1 s: What time do you want to travel? 2 u: Pardon? 3 s: Please state a departure time. 4 u: Five o’clock in the evening. 5 s: Is the departure time at seventeen hundred hours? 6 u: Yes.

Conversational Game Theory29 Parsing the Game Tree 1 s: What time do you want to travel? QW

Conversational Game Theory30 Parsing the Game Tree 2 u: Pardon? QW Pardon Unintelligible

Conversational Game Theory31 Parsing the Game Tree 3 s: Please state a departure time. PardonQW Unintelligible

Conversational Game Theory32 Parsing the Game Tree 4 u: Five o’clock in the evening. PardonQWReply Unintelligible

Conversational Game Theory33 Parsing the Game Tree 5 s: Is the departure time at seventeen hundred hours? PardonQWReplyConf Unintelligible

Conversational Game Theory34 Parsing the Game Tree 6 u: Yes. YesPardonQWReplyConf No Unintelligible

Conversational Game Theory35 Game Types and Move Types Game Types are sets of States and Move Types, and are operators on commitments. Move Types edges between states, can be either Game Types or Atomic Types, and are operators on propositions.

Conversational Game Theory36 Game Types Game TypeOperation Question addλ(p,q).[p  q] Pardon copyλ(p,q).[p] Information addλ(p,q).[p  q]

Conversational Game Theory37 Atomic Move Types Move TypeOperation Hello copyλ(p,q).p Reply-Yes promoteλ(p,q).[promote(p,q)] Reply-No deleteλ(p,q).[p - q]

Conversational Game Theory38 Realizing Games A game is realized with a preposition under discussion q. For the question game in the example, the question type was realized as the preposition travel_time(x) or “a query game about travel time”. A confirmation game might be realized with the preposition travel_time(17:00).

Conversational Game Theory39 Plans and Preferences Since games are routes toward committed propositions, plans can be made that are simply partially ordered stacks of games. Plans can be formed by Horn clause solvers, or other means. Preferences about how to choose and parse moves can be adjusted with probabilistic game tree parsing and high- level features.

Conversational Game Theory40 Conclusions - Pros Conversational Game Theory appears to be loosely based on Game Theory, with many added complications. It’s an interesting way to define the intentional structure of dialogue into a declarative compositional data structure. This intentional data structure can be computed over to generate and interpret dialogue, with high-level parameters that correlate with a theoretically sound notion of utility.

Conversational Game Theory41 Conclusions - Cons It seems very untested. There’s not much literature and even fewer working systems. The working systems are toys. So is it easy to develop more complex systems? Is it generic enough for a wide range of domains? Not everyone likes formal systems.

Conversational Game Theory42 Another Con A problem with logical omniscience. –K(t, p) == “t knows that p” –r => (p => q) == “r implies that p implies q” –[K(t, p) and K(t, r)] => K(t, q) ?? Always assumed in game theory, but even Sherlock Holmes fails this sometimes (and Watson fails often). Probably not a big deal; when introduced to p, r, and q, t will immediately accept q. There is research in psychology that may qualify the logic of the K proposition a little better.