Arpita rpita Bisw swas as PhD Stude udent nt (Google Fellow) - PowerPoint PPT Presentation

Speaker Arpita rpita Bisw swas as PhD Stude udent nt (Google Fellow) Game Theory y Lab, Dept. . of CSA, Indi dian n Institut stitute of Science nce, , Bangalore re arpita ta.bisw iswas@li as@live.in e.in Email address ress:

OUTLINE Game Theory – Basic Concepts and Results Mechanism Design Cooperative Game Theory Real-World Applications

GAME THEORY “Mathematical framework for rigorous study of conflict and cooperation among rati tiona nal and int ntellige gent nt agents”

GAME THEORY “Mathematical framework for rigorous study of conflict and cooperation among rati tiona nal and int ntellige gent nt agents” the agent would always choose an action that maximizes her/his (expected) utility. • competent enough to make any inferences about the game that a game theorist can make. • can carry out the required computations involved in determining a best response strategy

GAME THEORY “Mathematical framework for rigorous study of conflict and cooperation among rati tiona nal and int ntellige gent nt agents” the agent would always choose an action that maximizes her/his (expected) utility. preferences of the players expressed in terms of real numbers • competent enough to make any inferences about the game that a game theorist can make. • can carry out the required computations involved in determining a best response strategy

PRISONER’S DILEMMA The problem is as follows:  Two individuals arrested for a robbery (witnessed by several people).  The police suspects that they were guilty of a similar crime earlier, but were never caught.  The prisoners are lodged in separate prisons and interrogation happens separately  The police tells each prisoner that: a. “If you are the only one to confess, you’ll get a light sentence of 1 year while the other would be sentenced to 10 years in jail”. b. “If both of you confess, both of you would be sentenced for 5 years” . c. “If neither of you confess, then each of you would get 3 years in jail”.  The police also informs each prisoner that the same has been told to the other prisoner.

PRISONER’S DILEMMA Two Player ers: s: Bubly Confess ess Not ot Confess ess • Bunty Bunty ty • Bubly Two Ac Action ions: s: Confess ess • Confess • Not Confess Not ot Confess ess The utilit ity y matrix rix models the strategic conflict when two players have to choose their priorities

PRISONER’S DILEMMA Two Player ers: s: Bubly Confess ess Not ot Confess ess • Bunty Bunty ty • Bubly Two Ac Action ions: s: Confess ess • Confess • Not Confess < 𝑂, (𝐵 𝑗 ) 𝑗∈𝑂 , (𝑉 𝑗 ) 𝑗∈𝑂 > Not ot Confess ess 𝑂 ∶ set of players 𝐵 𝑗 ∶ set of actions for player 𝑗 𝑉 𝑗 ∶ 𝐵 1 × ⋯ × 𝐵 |𝑂| → ℝ The utilit ity y matrix rix models the strategic conflict when Action profile or two players have to choose their priorities Strategy profile

PRISONER’S DILEMMA Two Player ers: s: Bubly Confess ess Not ot Confess ess • Bunty Bunty ty • Bubly Two Ac Action ions: s: Confess ess • Confess • Not Confess Not ot Confess ess

PRISONER’S DILEMMA Two Player ers: s: Bubly Confess ess Not ot Confess ess • Bunty Bunty ty • Bubly Two Ac Action ions: s: Confess ess • Confess • Not Confess Not ot Confess ess

PRISONER’S DILEMMA Two Player ers: s: Bubly Confess ess Not ot Confess ess • Bunty Bunty ty • Bubly Two Ac Action ions: s: Confess ess • Confess • Not Confess Not ot Confess ess

PRISONER’S DILEMMA Two Player ers: s: Bubly Confess ess Not ot Confess ess • Bunty Bunty ty • Bubly Two Ac Action ions: s: Confess ess • Confess • Not Confess Not ot Confess ess

PRISONER’S DILEMMA Two Player ers: s: Bubly Confess ess Not ot Confess ess • Bunty Bunty ty • Bubly Two Ac Action ions: s: Confess ess • Confess • Not Confess Not ot Confess ess Nash h Equi Eq uilibr briu ium

PRISONER’S DILEMMA Two Player ers: s: Bubly Confess ess Not ot Confess ess • Bunty Bunty ty • Bubly Two Ac Action ions: s: Confess ess • Confess • Not Confess Not ot Confess ess Nash h Equi Eq uilibr briu ium A strategy profile in which no player gains by changing only his/her own strategy (assuming no one else changes their strategy)

PROJECT COORDINATION GAME Two Player ers: s: Bob Deep p Le Learn rning ng Webs bsite e Desi signi gning • Alice Alice • Bob Two Ac Action ions: s: Deep • Deep Learning Learning Project • Website Designing Project Websi site Designin gning

PROJECT COORDINATION GAME Two Player ers: s: Bob Deep p Le Learn rning ng Webs bsite e Desi signi gning • Alice Alice • Bob Two Ac Action ions: s: Deep • Deep Learning Learning Project • Website Designing Project Websi site Designin gning

PROJECT COORDINATION GAME Two Player ers: s: Bob Deep p Le Learn rning ng Webs bsite e Desi signi gning • Alice Alice • Bob Two Ac Action ions: s: Deep • Deep Learning Learning Project • Website Designing Project Websi site Designin gning Nash h Equil quilib ibria

PROJECT COORDINATION GAME Two Player ers: s: Bob Deep p Le Learn rning ng Webs bsite e Desi signi gning • Alice Alice • Bob Two Ac Action ions: s: Deep • Deep Learning Learning Project • Website Designing Project Websi site Designin gning Nash h Equil quilib ibria Does s there exist st there e any ot other er Nash h Equi quilibr ibrium ium in this s game? e?

PROJECT COORDINATION GAME Two Player ers: s: Bob Deep p Le Learn rning ng Webs bsite e Desi signi gning • Alice Alice • Bob Two Ac Action ions: s: Deep • Deep Learning Learning Project (DL) • Website Designing Project (WD) Websi site Designin gning Nash h Equil quilib ibria Does s there exist st there e any ot other er Nash h Equi quilibr ibrium ium in this s game? e? Alice: With probability 2/3 choose DL and with probability 1/3 choose WD Bob: With probability 1/3 choose DL and with probability 2/3 choose WD

MIXED STRATEGY NASH EQUILIBRIUM

EXISTENCE OF NASH EQUILIBRIA IN GAMES  Does Nash Equilibrium always exist?

EXISTENCE OF NASH EQUILIBRIA IN GAMES  Does Nash Equilibrium always exist? 𝑭𝒘𝒇𝒔𝒛 𝒈𝒋𝒐𝒋𝒖𝒇 𝒕𝒖𝒔𝒃𝒖𝒇𝒉𝒋𝒅 𝒈𝒑𝒔𝒏 𝒉𝒃𝒏𝒇 𝒊𝒃𝒕 𝒃𝒖 𝒎𝒇𝒃𝒕𝒖 ,𝑶𝒃𝒕𝒊 𝑼𝒊𝒇𝒑𝒔𝒇𝒏, 𝟐𝟘𝟔𝟏-. 𝒑𝒐𝒇 𝒏𝒋𝒚𝒇𝒆 𝒕𝒖𝒔𝒃𝒖𝒇𝒉𝒛 𝑶𝒃𝒕𝒊 𝑭𝒓𝒗𝒋𝒎𝒋𝒄𝒔𝒋𝒗𝒏.

EXISTENCE OF NASH EQUILIBRIA IN GAMES  Does Nash Equilibrium always exist? 𝑭𝒘𝒇𝒔𝒛 𝒈𝒋𝒐𝒋𝒖𝒇 𝒕𝒖𝒔𝒃𝒖𝒇𝒉𝒋𝒅 𝒈𝒑𝒔𝒏 𝒉𝒃𝒏𝒇 𝒊𝒃𝒕 𝒃𝒖 𝒎𝒇𝒃𝒕𝒖 ,𝑶𝒃𝒕𝒊 𝑼𝒊𝒇𝒑𝒔𝒇𝒏, 𝟐𝟘𝟔𝟏-. 𝒑𝒐𝒇 𝒏𝒋𝒚𝒇𝒆 𝒕𝒖𝒔𝒃𝒖𝒇𝒉𝒛 𝑶𝒃𝒕𝒊 𝑭𝒓𝒗𝒋𝒎𝒋𝒄𝒔𝒋𝒗𝒏.  Is there an efficient algorithm for computing a mixed Nash equilibrium?

EXISTENCE OF NASH EQUILIBRIA IN GAMES  Does Nash Equilibrium always exist? 𝑭𝒘𝒇𝒔𝒛 𝒈𝒋𝒐𝒋𝒖𝒇 𝒕𝒖𝒔𝒃𝒖𝒇𝒉𝒋𝒅 𝒈𝒑𝒔𝒏 𝒉𝒃𝒏𝒇 𝒊𝒃𝒕 𝒃𝒖 𝒎𝒇𝒃𝒕𝒖 ,𝑶𝒃𝒕𝒊 𝑼𝒊𝒇𝒑𝒔𝒇𝒏, 𝟐𝟘𝟔𝟏-. 𝒑𝒐𝒇 𝒏𝒋𝒚𝒇𝒆 𝒕𝒖𝒔𝒃𝒖𝒇𝒉𝒛 𝑶𝒃𝒕𝒊 𝑭𝒓𝒗𝒋𝒎𝒋𝒄𝒔𝒋𝒗𝒏.  Is there an efficient algorithm for computing a mixed Nash equilibrium? 𝑮𝒋𝒐𝒆𝒋𝒐𝒉 𝒏𝒋𝒚𝒇𝒆 𝒕𝒖𝒔𝒃𝒖𝒇𝒉𝒛 𝑶𝒃𝒕𝒊 𝑭𝒓𝒗𝒋𝒎𝒋𝒄𝒔𝒋𝒗𝒏 ,𝑬𝒃𝒕𝒍𝒃𝒎𝒃𝒍𝒋𝒕 𝒇𝒖 𝒃𝒎. , 𝟑𝟏𝟏𝟕-. 𝒋𝒕 𝑸𝑸𝑩𝑬 − 𝒅𝒑𝒏𝒒𝒎𝒇𝒖𝒇

OTHER TYPES OF EQUILIBRIA  Strongly gly Dominant inant Strategy egy Equi quilibr ibriu ium m (SDSE SE): ): 𝐵𝑜 𝑏𝑑𝑢𝑗𝑝𝑜 𝑞𝑠𝑝𝑔𝑗𝑚𝑓 𝑏 1 , ⋯ , 𝑏 𝑜 𝑗𝑡 𝑑𝑏𝑚𝑚𝑓𝑒 𝒕𝒖𝒔𝒑𝒐𝒉𝒎𝒛 𝒆𝒑𝒏𝒋𝒐𝒃𝒐𝒖 𝒕𝒖𝒔𝒃𝒖𝒇𝒉𝒛 𝒇𝒓𝒗𝒋𝒎𝒋𝒄𝒔𝒋𝒗𝒏 𝑔𝑝𝑠 𝑏 𝑕𝑏𝑛𝑓 < 𝑂, 𝐵 𝑗 , 𝑉 𝑗 >, 𝑗𝑔 ∀𝑗 ∈ 𝑂 𝑏𝑜𝑒 ∀𝑐 𝑗 ∈ 𝐵 𝑗 ∖ *𝑏 𝑗 + , 𝑉 𝑗 𝑏 𝑗 , 𝑐 −𝑗 > 𝑉 𝑗 𝑐 𝑗 , 𝑐 −𝑗 ∀𝑐 −𝑗 ∈ 𝐵 −𝑗 . Bubly Confess ess Not ot Confess ess Bunty ty Confess ess Not ot Confess ess