ECE700.07: Game Theory with Engineering Applications Le Lecture 7: - PowerPoint PPT Presentation

ECE700.07: Game Theory with Engineering Applications Le Lecture 7: Games with Incomplete Information Seyed Majid Zahedi

Outline • Bayesian games • Bayes-Nash equilibrium • Auctions • Readings: • MAS Sec. 6.3, GT Sec. 6.1 - 6.5

Bayesian Games: Games of Incomplete Information • So far, we assumed all agents know what game they are playing • Number of agents • Actions available to each agent • Utilities associated with each outcome • In extensive form games, actions may not be common knowledge, but game itself is • Bayesian games allow us to represent agents’ uncertainties about game being played • Uncertainty is represented as commonly known probability distribution over possible games • We make following assumptions • All games have same number of agents and same strategy space for each agents • Possible games only differ in agents’ utilities for each outcome • Beliefs are posteriors, obtained by conditioning common prior on private signals

Example: Bayesian Entry Deterrence Game • Incumbent decides whether to build new plant, entrant decides whether to enter • Incumbent knows her cost, entrant is uncertain if incumbent’s building cost is 4 or 1 • Game takes one of following two forms Entrant Entrant Enter Stay Out Enter Stay Out Incumbent Incumbent Build (0, -1) (2, 0) Build (3, -1) (5, 0) Don’t Build (2, 1) (3, 0) Don’t Build (2, 1) (3, 0) High Building Cost Low Building Cost • Suppose entrant assigns prior probability of 𝑞 to incumbent’s cost being high • Incumbent’s dominant strategy is “build” if cost is low and “don’t build” otherwise • Entrant’s utility is 2𝑞 − 1 for “enter” and and 0 for “stay out” • Entrant enters if 𝑞 > 1/2

<latexit sha1_base64="jsogXH6ITD7uPZegRi/xl1Nva1E=">ACEXicbZDLSgMxFIYzXmu9jbp0EyxCRxnevECIgU3LqvYC7SlZNK0Dc1khiSjDENfwY2v4saFIm7dufNtTKdF1PpD4OM/53ByfjdgVCrb/jRmZufmFxZTS+nldW1dXNjsyr9UGBSwT7zRd1FkjDKSUVRxUg9EAR5LiM1d3AxqtduiZDU5zcqCkjLQz1OuxQjpa2mXWsYrSft4pZ5yDaO89pTqh5TXt9hYTw72B05hzm2mbGtuxEcBqcCWTAROW2+dHs+Dj0CFeYISkbjh2oVoyEopiRYboZShIgPEA90tDIkUdkK04uGsJd7XRg1xf6cQUT9+dEjDwpI8/VnR5Sfm3NjL/qzVC1T1pxZQHoSIcjxd1QwaVD0fxwA4VBCsWaUBYUP1XiPtIKx0iOkhNORjr5PnoZqznLyVuGqkCkVJ3GkwDbYAVngGNQApegDCoAg3vwCJ7Bi/FgPBmvxtu4dcaYzGyBXzLevwBXMZm3</latexit> Example: Bayesian Entry Deterrence Game (cont.) • Now suppose entrant is uncertain if incumbent’s building cost is 4 or 2.5 Entrant Entrant Enter Stay Out Enter Stay Out Incumbent Incumbent Build (0, -1) (2, 0) Build (1.5, -1) (3.5, 0) Don’t Build (2, 1) (3, 0) Don’t Build (2, 1) (3, 0) High Building Cost Low Building Cost • “Don’t build” is still dominant strategy for incumbent if cost is high • Incumbent’s strategy if cost is low depends on her prediction of entrant’s strategy • If 𝑧 is incumbent’s prediction of entrant playing “enter”, then “build” is better if 1 . 5 y + 3 . 5(1 − y ) > 2 y + 3(1 − y ) ⇒ y < 1 / 2 • Incumbent must predict entrant's strategy • Entrant cannot infer incumbent’s strategy only from her knowledge of utilities

Example: Bayesian Entry Deterrence Game (cont.) Nature • We can model game as extensive form game 𝑞 1 − 𝑞 High Cost Low Cost • Nature chooses incumbent’s type Incumbent • Agents have same prior belief about nature’s move Build Don’t Build Don’t • Suppose that Entrant • Incumbent chooses build with probability 𝑦 if cost is low • Entrant chooses enter with probability 𝑧 Enter Don’t Enter Don’t Enter Don’t Enter Don’t • What is incumbent's best response to 𝑧 if cost is low? • 𝑦 = 1 if 𝑧 < 1/2 and 𝑦 = 0 if 𝑧 > 1/2 (2,0) (0,-1) (2,1) (3,0) (1.5,-1) (3.5,0) (2,1) (3,0) • 𝑦 ∈ [0,1] if 𝑧 = 1/2 • What is entrant’s best response to 𝑦 ? • 𝑧 = 1 if 𝑦 < 1/2 1 − 𝑞 and 𝑧 = 0 if 𝑦 > 1/2 1 − 𝑞 • 𝑧 ∈ [0,1] if 𝑦 = 1/2 1 − 𝑞 • Search for Bayes-Nash equilibrium boils down to finding 𝑦, 𝑧 that are optimal for both 0,1 for any 𝑞 or 1,0 if and only if 𝑞 ≤ 1/2 or 1/2 1 − 𝑞 , 1/2 •

Bayesian Games Model • Bayesian game is tuple ⟨ℐ, 𝑇 4 4∈ℐ , Θ 4 4∈ℐ , 𝑞, 𝑣 4 4∈ℐ ⟩ • ℐ is finite set of agents • 𝑇 4 is set of actions available to agent 𝑗 • Θ 4 is type space of agent 𝑗 • 𝑞: Θ ⟼ 0,1 is common prior over types • 𝑣 4 : 𝑇×Θ ⟼ ℝ is utility function for agent 𝑗 • Agent 𝑗 ’s mixed strategy 𝜏 4 : Θ 4 → Σ 4 is contingency plan for all 𝜄 4 ∈ Θ 4 • 𝜏 4 𝜄 4 specifies 𝑗 ’s mixed strategy when her type is 𝜄 4 • 𝜏 4 𝑡 4 𝜄 4 specifies probability of agent 𝑗 taking action 𝑡 4 when her type is 𝜄 4

<latexit sha1_base64="1LpV7+rAkz2PiP9GnPuTeU6g5s=">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</latexit> <latexit sha1_base64="kvXERXAaJOmchKdNGX58DBQmdoE=">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</latexit> <latexit sha1_base64="xdvYnTUZRLXk27ExJHcNZkbM2Ok=">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</latexit> Expected Utilities • Ex-post expected utility Y ! X EU i ( σ , θ ) = σ j ( s j | θ j ) u i ( s, θ ) s ∈ S j ∈ I • Ex-interim expected utility X EU i ( σ , θ i ) = p ( θ − i | θ i ) EU i ( σ , ( θ i , θ − i )) θ − i ∈ Θ − i • Ex-ante expected utility X X EU i ( σ ) = p ( θ i ) EU i ( σ , θ i ) = p ( θ ) EU i ( σ , θ ) θ i ∈ Θ i θ ∈ Θ

Example • Consider following game which consists of four 2 × 2 games • Matching Pennies, Prisoner’s Dilemma, Coordination and Battle of the Sexes 𝜄 CB 𝜄 CC MP PD 2,0 0,2 2,2 0,3 𝜄 BB 0,2 2,0 3,0 1,1 𝑞 = 0.3 𝑞 = 0.1 Coord BoS 2,2 0,0 2,1 0,0 𝜄 BC 0,0 1,1 0,0 1,2 𝑞 = 0.2 𝑞 = 0.4

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