Conversational Game Theory

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Conversational Game Theory

• Thomas K Harris
• Graduate Seminar on Dialog Processing
• November 25, 2003

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• 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
• Lewin _at_ SRI Cambridge later with the EUs
  TRINDI
• Some Evaluation

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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.

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Game Theory Intro
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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. 1928 and 1937.
• Little interest until the publication of von
  Neumann and Morgensterns Theory of Games and
  Economic Behavior 1944.

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Would you like to play Thermonuclear War?

• c. 1950s 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 -gt Truce
• A beautiful mind expands the theory from
  competitive to collaborative games.

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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, Theres a related course on this here
  (Philosophy Dept) 80-305 Rational Choice

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Can you consistently order your alternatives?

• A preference ordering ? exists between any two
  outcomes, and it is transitive.

?
?
?
?
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Are you indifferent to compound lotteries?

• Compound lotteries can be reduced to simple
  lotteries.
• ½ ½ ( ½ ½ )
• ½ ¼ ¼

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Are Your Preferences Continuous?

• Each outcome Ai is indifferent to some lottery
  ticket involving just A1 and Ar, where for each
  Ai, A1 ? Ai and Ai ? Ar.
• 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 10 .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. Whats
  your p?

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Are you indifferent to prize substitutions?

• If youve 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.

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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
  )

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Is more of a good thing always better?

• Lotteries are monotonic.
• Assuming ?
• (p1 , (1-p1) ) ? (p2 , (1-p2)
  )
• if and only if p1 gt p2

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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.

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Irrational Agents

• What about fewer assumptions?
• Mathematical intractability unsolvable
  solutions ambiguous results.
• Still can be good science, more apt to be called
  psychology.

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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. viruss or other
  simple automata.

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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

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Games

• Chess 2-player, perfect information, zero-sum
• Bridge 2-player!, risk, zero-sum
• Rock-Paper-Scissors 2-player, perfect-information
  , zero-sum
• Prisoners dilemma 2-player, perfect-information,
  non-zero-sum

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A Common Game Tree
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Conversational Games
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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.

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Conversational Game Types

• Question Game

pardon
QW
RW
confirmation
QW-R
interrupt
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Conversational Game Types

• Pardon Game

Unrecognized
Pardon
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Conversational Game Types

• Confirmation Game

Explicit Confirmation
Mod
Yes
No
Implicit Confirmation
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Conversational Game Types

• Interruption Game

Unimportant
information
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Conversational Game Types

• Information Game

Information
confirmation
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Conversational Game Types

• Hello Game

Hello
Hello
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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 oclock in the evening.
• 5 s Is the departure time at seventeen hundred
  hours?
• 6 u Yes.

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Parsing the Game Tree

• 1 s What time do you want to travel?

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Parsing the Game Tree

• 2 u Pardon?

Unintelligible
Pardon
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Parsing the Game Tree

• 3 s Please state a departure time.

Unintelligible
Pardon
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Parsing the Game Tree

• 4 u Five oclock in the evening.

Unintelligible
Pardon
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Parsing the Game Tree

• 5 s Is the departure time at seventeen hundred
  hours?

Unintelligible
Pardon
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Parsing the Game Tree

• 6 u Yes.

Unintelligible
No
Yes
Pardon
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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.

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Game Types
Game Type Operation
Question add ?(p,q).p?q
Pardon copy ?(p,q).p
Information add ?(p,q).p?q
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Atomic Move Types
Move Type Operation
Hello copy ?(p,q).p
Reply-Yes promote ?(p,q).promote(p,q)
Reply-No delete ?(p,q).p - q
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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(1700).

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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.

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Conclusions - Pros

• Conversational Game Theory appears to be loosely
  based on Game Theory, with many added
  complications.
• Its 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.

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Conclusions - Cons

• It seems very untested. Theres 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.

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Another Con

• A problem with logical omniscience.
• K(t, p) t knows that p
• r gt (p gt q) r implies that p implies q
• K(t, p) and K(t, r) gt 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.