On Game Mechanisms and Procedural Fairness

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On Game Mechanisms and Procedural Fairness

• Moshe Looks
• Ronald P. Loui
• Washington University in St. Louis, USA

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Fairness

• Distributive Fairness (who gets what?)
• Procedures
• Can determine distributions
• Can also determine predicates
• Procedural vs. Substantive Fairness
• Substantive refers to outcomes (equal split)
• Procedural refers to construction of outcomes

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Procedural Fairness

• Procedure justifies outcome
• If properly conducted
• Parties contribute inputs
• Outputs are constructed upon inputschance
• Justify procedure independently
• Disputants
• Agree to play a game
• Bound to outcome of game
• Like contracts
• Society prescribes procedures

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Fairness as Appropriateness

• Structural Fairness
• Symmetry of rules
• Equality of opportunity
• Exchangeability of endowments resources
• Teleological Fairness
• Justifiable asymmetry
• Based on social roles
• Ultimatum game
• Let's flip a coin . But it's MY envelope Let's
  flip a biased coin
• Rawls role ex-ante exchangeability vs.
  positional ex-ante exchangeability

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Procedural Fairness

• Personally at stake
• Procedures for argument need to be
• Fair
• Efficient
• Appropriate
• I, Tom Gordon, Gerard Vreeswijk, Arno Lodder, and
  Rohit Parikh believe
• Correctness of computation fairness of
  procedure
• Past student efforts
• Anne Jump, Alan Bisarya, Ricky Wofsey, James
  Rosen, Alex Schiller

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What Formalism?

• JKP! (jon-ken-po/rock-paper-scissors) is
• One play of simultaneous choice from J, K, P
• ! Check for winner (possibly indeterminate)
• JKP!n
• As many as n rounds of JKP until winner
• More decisive than JKPn-1
• JKP!
• As many rounds as needed
• Uncountably decisive

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More Formalism

• HEADS is
• One play of INCA/.5 INCB/.5 HEADS
• HEADSkBEST! is best of k rounds of HEADS
• pkBEST! reduces effect of error/chance in p!
• Two counferfactuals
• If p has an A-bias, the probability of pkBEST! ?
  A
• pkBEST! is more like pBEST! than pk-1BEST!
• Larger k takes more time, can spread
  distributions

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More Formalism

• Chance determination of procedure
• CHANCE1 p pA/.5 pB/.5
• Determines by chance who moves first
• Effect of chance depends on later choice
• CHANCE1 NAMEHIGHESTNUMBER!
• CHANCE1 TICTACTOE!
• CHANCE1 CHESS!
• CHANCE1 GO!
• Also Chance determination of endowments
• CHANCEA CHANCEB MONOPOLY
• Endowment difference reducible to procedural
  choice
• Except when claiming strategic-familial
  resemblance

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More Formalism

• Baseball Innings
• INNA BATSA BATSB
• Apparent game is INNA9BEST!
• Actual game is INNA9BEST! XTRAA
• XTRAA (INNABEST!)
• Chance/Choice ontology is unclear with games of
  physical skill
• (derivably possibly) Fairer Procedures (no home
  info advantage)
• INNA9BEST! CHANCE1 XTRA
• CHANCE1(INN9BEST! XTRA)
• (CHANCE1 INN)9! (CHANCE1 INN BEST!)
• MERIT1(INN9BEST! XTRA)

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Q2 and Other Arguments

• Nested quantifiers (Q2)
• P is solvable (for A) iff
• there exists a sA s.t. for every sB, A wins
• TICTACTOE is solvable for 5 yr olds
• TICTACTOE is unsolvable for many
• Reject many dominant strategy arguments
• Players not omniscient, non-optimal
• Optimal play depends on population/opponent
• sA ??A
• Some reductions
• x CHANCEWIN! CHANCEWIN!

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Aspects with Formalizable Tradeoffs

• MERIT Degree to which rules promote a skill
• Management of risk
• Tactical lookahead
• CHANCE
• Meritocracy of good luck
• DECISIVENESS/EFFICIENCY
• Amount of input required
• Potential to be indeterminate
• SPREAD
• Range of outcomes

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Future Axioms

• CHANCE1!
• Removes specific asymmetry
• Usually increases ex-ante exchangeability
• Usually decreases merit, ex-post exchangeability
• pkBEST!
• Magnifies meritocracy of p!
• Decreases decisiveness/efficiency
• JKP!
• Functions like an unbiased chance mechanism

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Prospects

• Main technical problem was notation for
  procedures
• Main problem now is that the subject is large and
  unexplored
• Trying to make simple self-contained observations
  is challenging
• Building inventory of procedural primitives
• We are at a naïve stage
• Rawl's "pure procedural justice" and "veil of
  ignorance" welfare arguments DO seem naïve to us!