Game Theoretic Approach in Computer Science CS3150, Fall 2002 Mechanism Design 2

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Game Theoretic Approach in Computer
ScienceCS3150, Fall 2002Mechanism Design (2)

• Patchrawat Uthaisombut
• University of Pittsburgh

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Topics

• Consequences
• Preferences
• Alternate definitions of Strategic Games
• Choice rules / choice functions
• Game forms
• Environment

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Consequences

• C denotes the set of consequences or outcomes
• Example
• o1 Apichai and Buncha meet at Pungkang
• o2 Apichai and Buncha meet at Samkok
• o3 Apichai and Buncha do not meet.

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Preference Relations

• A preference relation is
• a complete reflexive transitive binary relation
• Complete
• For all a,b in X a gt b or b gt a
• Reflexive
• For all a in X a gt a
• Transitive
• For all a,b,c in X a gt b and b gt c implies a gt c

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Preferences

• Preference of player i
• x gti y
• player i prefers outcome x to outcome y
• For player i, outcome x is as good as or better
  than y
• defined over C (set of all consequences)
• defined over A (set of all action profiles)
• More general than utility functions ui
• Example
• o1 gtA o2 gtA o3
• o2 gtB o1 gtB o3
• (p,p) gtA (s,s) gtA (p,s) A (s,p)

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Strategic Games with Utility Function

• A strategic game is a 3-tuple (n,A,u)
• The number of players n.
• For 1ltiltn, a set Ai of actions for player i.
• For 1ltiltn, a payoff function uiA1??An ? R for
  player i.

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Strategic Games with Utility Function

• Players n 2
• Player 1 Apichai
• Player 2 Buncha
• Actions
• A1 pungkang, samkok
• A2 pungkang, samkok
• Payoffs
• u1(pungkang,pungkang ) 2
• u1(pungkang,samkok ) 0
• u1(samkok,pungkang ) 0
• u1(samkok,samkok ) 1

• u2(pungkang,pungkang ) 1
• u2(pungkang,samkok ) 0
• u2(samkok,pungkang ) 0
• u2(samkok,samkok ) 2

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Strategic Games General Form

• A strategic game is a 3-tuple (n,A,gt)
• The number of players n.
• For 1ltiltn, a set Ai of actions for player i.
• For 1ltiltn, a preference relation gti on A for
  player i.

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Strategic Games General From

• Players n 2
• Player 1 Apichai
• Player 2 Buncha
• Actions
• A1 pungkang, samkok
• A2 pungkang, samkok
• Preferences
• (p,p) gt1 (s,s) gt1 (s,p) 1 (p,s)
• This also implies that
• (p,p) gt1 (s,p), (p,p) gt1 (p,s)
• (s,s) gt1 (p,s)

(s,s) gt2 (p,p) gt2 (s,p) 2 (p,s) This also
implies that (s,s) gt2 (s,p), (s,s) gt2
(p,s) (p,p) gt2 (p,s)
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Strategic Game with Outcomes

• A strategic game is a 5-tuple (n,A,C,g,gt)
• The number of players n.
• For 1ltiltn, a set Ai of actions for player i.
• Set of consequences C
• Outcome function gA?C
• For 1ltiltn, a preference relation gti on C

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Strategic Games with Outcomes

• Players n 2
• Player 1 Apichai
• Player 2 Buncha
• Actions
• A1 pungkang, samkok
• A2 pungkang, samkok
• Consequences
• C o1, o2, o3
• o1 they meet at PK
• o2 they meet at SK
• o3 they do not meet

• Outcome function
• g(p,p) o1
• g(s,s) o2
• g(p,s) o3
• g(s,p) o3
• Preferences
• o1 gt1 o2 gt1 o3
• This also implies that
• o1 gt1 o3
• o2 gt2 o1 gt2 o3
• This also implies that
• o2 gt2 o3

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Choice Rules / Choice Functions

• Goal of the principal/game designer
• fP?2C
• fU?2C
• Given a preference profile of the players, f
  specifies the outcome(s) the principal wants.

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What has to be designed?

• What exactly does a planner has to design?
• Games?
• (N,A,u), (N,A,gt)
• The preference relation is not yet specified
  during design time.
• gt is unknown to the planner.
• Game forms with consequences in C
• (N, A, g)
• Outcome function gA?C

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Games and Game Forms

• Game form (N, A, g) with consequences in C and
  preference profile gt over C induces a game
  (n,A,C,g,gt), or equivalently (N,A,gt)
• gt is defined over A (set of all action profiles)
• For a,b in A, a gt b if and only if g(a) gt g(b)

(N, A, g) in C
gt
(N, A, C, g, gt)


(N, A, gt)

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Environment

• What are available to the planner?
• The environment consists of
• a finite set N of players, with Ngt2
• a set C of outcomes
• a set P of preference profiles over C
• a set G of game forms with consequences in C

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Environment in Single-Item Auction

• N
• C set of (k,(ci))
• P set of gt set of u set of (vi)
• G set of (N,(Ai),g)
• Planners goal
• fP?2C or fU?2C