offer Mr. Vieweg from Berlin an epic poem, Hermann and Dorothea,... Concerning the royalty we will proceed as follows: I will hand over to Mr. Counsel Böttiger a sealed note which contains my demand, and I wait for what Mr. Vieweg will suggest to offer for my work. If his offer is lower than my demand, then I take my note back, unopened, and the negotiation is broken. If, however, his offer is higher, then I will not ask for more than what is written in the note to be opened by Mr. Böttiger. Letter from Goethe to publisher, 1797
wins Pays amount second highest bidder offered 6 For Goethe’s “auction” – there is only one real bidder (Vieweg publisher). Goethe is acting as the second bidder (but more like a “reserve price”).
It is this: the publisher always knows the profit ..., whereas the author is totally in the dark.’’ (Goethe’s letter) Reduce information asymmetry between Goethe and publisher
offer Mr. Vieweg from Berlin an epic poem, Hermann and Dorothea,... Concerning the royalty we will proceed as follows: I will hand over to Mr. Counsel Böttiger a sealed note which contains my demand, and I wait for what Mr. Vieweg will suggest to offer for my work. If his offer is lower than my demand, then I take my note back, unopened, and the negotiation is broken. If, however, his offer is higher, then I will not ask for more than what is written in the note to be opened by Mr. Böttiger. Letter from Goethe to publisher, 1797 Did it work out for Goethe?
an epic poem, Hermann and Dorothea,... Concerning the royalty we will proceed as follows: I will hand over to Mr. Counsel Böttiger a sealed note which contains my demand, and I wait for what Mr. Vieweg will suggest to offer for my work. If his offer is lower than my demand, then I take my note back, unopened, and the negotiation is broken. If, however, his offer is higher, then I will not ask for more than what is written in the note to be opened by Mr. Böttiger. Letter from Goethe to publisher, 1797 The sealed note with the imprisoned Golden Wolf is really in my office. Now, tell me what can and will you pay? I put myself in your place, dear Vieweg, and feel what a spectator, who is your friend, can feel. Given what I approximately know about Goethe’s fees from Göschen, Bertuch, Cotta and Unger, let me just add one thing: you cannot bid under 200 Friedrichs d’or. Böttiger’s letter to Vieweg
really in my office. Now, tell me what can and will you pay? I put myself in your place, dear Vieweg, and feel what a spectator, who is your friend, can feel. Given what I approximately know about Goethe’s fees from Göschen, Bertuch, Cotta and Unger, let me just add one thing: you cannot bid under 200 Friedrichs d’or. Böttiger’s letter to Vieweg 1 F d’or: ~6 grams of gold = $255 today 200 x 255 = $51,000 (~10 J. K. Rowling’s advance for first Harry Potter + 15% royalty ~ $)
. ∈1 set of players . action space for player . : → ℝ utility function for player Complete Information Game . = () = (, ) = argmax. ∈1 .. (highbidder wins) = J , argmaxK ∈ 1 L {.} K (pays second price) Note: not dealing with ties here
. ∈1 . = () = (, ) = argmax. ∈1 .. (highbidder wins) = J , argmaxK ∈ 1 L {.} K (pays second price) Note: not dealing with ties here . = value of item to player . = Z . − if = 0 if ≠
. ∈1 set of players . action space for player . : → ℝ utility function for player Complete Information Game If all utilities are known, no need for the auction! Seller should just sell to buyer with highest utility at that price.
. ∈1 set of players . action space for player . : → ℝ utility function for player Complete Information Game = , . , . , . . ∈1 , . type (of player) space distribution over types . ∈ . type of player (not publicly known) . : , → ℝ utility function for player by type. Incomplete Information Game
. . ∈1 , . = () = (, ) = argmax. ∈1 .. = J , argmaxK ∈ 1 L {.} K . = value of item to player unknown to others . (. , ) = Z . − if = 0 if ≠ . = {. ∈ ℝg} possible different values
is independent (no collusion) • is selfish (only cares if she wins) • is greedy (wants to pay lowest price to win) • prefers to win if she can pay ≤ value of good to her: . Based on Giacomo Bonanno, Game Theory (free on-line PDF) Partial ordering of outcomes:
is independent (no collusion) • is selfish (only cares if she wins) • is greedy (wants to pay lowest price to win) • prefers to win if she can pay ≤ value of good to her: . Based on Giacomo Bonanno, Game Theory (free on-line PDF) Partial ordering of outcomes: , ≻. , l iff < l , ≻. , l for all ≠ , ≤ .
players, it is a weakly dominant strategy for player to bid her true value, . = .. 22 Proof: Case 1: . < .. Recall partial ordering of outcomes: , ≻. , l iff < l , ≻. , l for all ≠ , ≤ .
players, it is a weakly dominant strategy for player to bid her true value, . = .. 23 Proof: Case 1: . < .. Recall partial ordering of outcomes: , ≻. , l iff < l , ≻. , l for all ≠ , ≤ . If outcome is , l , l > . : can’t win, without paying more than . . If outcome is , l , ′ ≤ . : increasing . to . > ′ > . improves outcome since , ≻. , l for all ≠ , ≤ . If outcome is , , since =J , argmaxK ∈ 1 L {.} K increasing . does not change outcome.
players, it is a weakly dominant strategy for player to bid her true value, . = .. 24 Proof: Case 2: . > .. Recall partial ordering of outcomes: , ≻. , l iff < l , ≻. , l for all ≠ , ≤ . If outcome is , l , l > . : can’t win, without paying more than . . If outcome is , : if > . this has negative utility for : improves by lowering bid to lose if ≤ ., makes no difference if . is lowered to .. . (. , ) = Z . − if = 0 if ≠
players, it is a weakly dominant strategy for player to bid her true value, . = .. 25 Proof: Case 1: . < . . either no change or improve outcome by increasing .to .. Case 2: . > .. either no change or improve outcome by decreasing .to .. Recall partial ordering of outcomes: , ≻. , l iff < l , ≻. , l for all ≠ , ≤ . . (. , ) = Z . − if = 0 if ≠ Hence, . > . weakly dominates any other strategy.
players, it is a weakly dominant strategy for player to bid her true value, . = .. 26 Recall partial ordering of outcomes: , ≻. , l iff < l , ≻. , l for all ≠ , ≤ . . (. , ) = Z . − if = 0 if ≠ Hence, . > . weakly dominates any other strategy. When isn’t this true?
> Queen = 2 players, each player gets one card face-up . = { bet, fold } ℎ = K = bet, ∈ } = argmax. ∈.uvwux . = | ℎ | All players in hand bet 1 . (, ) = { − 1 if = 0 if ≠ and . = fold −1 if ≠ and . = bet . = {
gets one card face-down (only player can view) . = { bet, fold } = { } Player sees only . Full state of the game is ⋃ . . . (, ) = { − 1 if = 0 if ≠ and . = fold −1 if ≠ and . = bet Is this an incomplete information game?
are known to everyone Incomplete Information Game: do not have full knowledge of other players utility functions Perfect Information Game: full state of the world completely known to all players Imperfect Information Game: players only have partial knowledge of information state 30
one card face down Betting: (“half street” game) Ante: 1 chip Player 1: bet 1, or check Player 2: call or fold Loosely based on Bill Chen and Jerrod Ankenman, The Mathematics of Poker.
> Queen = 2 players, each player gets one card face-up . = { bet, fold } = { . } ℎ = K = bet, ∈ } = argmax. ∈.uvwux . = | ℎ | All players in hand bet 1 . (, ) = { − 1 if = 0 if ≠ and . = fold −1 if ≠ and . = bet