MARKETS, MECHANISMS, MACHINES University of Virginia, Spring 2019 Class 12: Imperfect Information Games 21 February 2019 cs4501/econ4559 Spring 2019 David Evans and Denis Nekipelov https://uvammm.github.io

Plan Incomplete Information Games Second Price Auctions Imperfect Information Games Puny Poker 1

Second Price Auction Bidders submit sealed bids Highest bidder wins Pays amount second highest bidder offered 2

3 Johann Wolfgang von Goethe (1749-1832)

4 Johann Wolfgang von Goethe (1749-1832)

5 Johann Wolfgang von Goethe (1749-1832) I am inclined to 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

Example: Second Price Auction Bidders submit sealed bids Highest bidder 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”).

Who wants a second price auction? When it is better for the buyer than a first price auction? 7 When it is better for the seller than a first price auction?

8 ‘‘Let me . . . name the main evil. 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

9 Johann Wolfgang von Goethe (1749-1832) I am inclined to 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?

10 I am inclined to 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 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

11 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 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 ~ $)

Formalizing Second Price Auction 12 = , . , . . ∈1 set of players . action space for player . : → ℝ utility function for player Complete Information Game

Formalizing Second Price Auction 13 = , . , . . ∈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

Formalizing Second Price Auction 14 = , . , . . ∈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 ≠

Formalizing Second Price Auction 15 = , . , . . ∈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.

Formalizing Second Price Auction 16 = , . , . . ∈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

Second Price Auction 17 = , . , . , . . ∈1 , . = () = (, ) = argmax. ∈1 .. = J , argmaxK ∈ 1 L {.} K

Second Price Auction 18 = , . , . , . . ∈1 , . = () = (, ) = argmax. ∈1 .. = J , argmaxK ∈ 1 L {.} K . = value of item to player unknown to others . (. , ) = Z . − if = 0 if ≠ . = {. ∈ ℝg} possible different values

Second Price Auction: Optimal Strategy 19 Assumptions: Each player: • 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:

Second Price Auction: Optimal Strategy 20 Assumptions: Each player: • 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 ≠ , ≤ .

Vickrey’s Theorem (1961) In a second-price auction of selfish and greedy players, it is a weakly dominant strategy for player to bid her true value, . = .. 21

Vickrey’s Theorem In a second-price auction of selfish and greedy 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 ≠ , ≤ .

Vickrey’s Theorem In a second-price auction of selfish and greedy 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.

Vickrey’s Theorem In a second-price auction of selfish and greedy 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 ≠

Vickrey’s Theorem In a second-price auction of selfish and greedy 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.

Vickrey’s Theorem In a second-price auction of selfish and greedy 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?

A+K+Q Puny Poker Flickr:cc Malkav

Pico Poker: Game Rules 3 card deck: Ace > King > 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 . = {

Pico Poker: Adding Down Cards = 2 players, each player 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?

Definitions Summary Complete Information Game: utility functions of all players 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

Pico Poker: Optimal Strategy? = 2 players, each player gets one card face-down (only player can view) . ∈ A, K, Q , ≠ ⟹ . ≠ K (uniformly random) . = { bet, fold } . = { . } . (, ) = { − 1 if = 0 if ≠ , . = fold −1 if ≠ , . = bet

Sequential Games Players take turns moving, see history of previous moves 32 A pure strategy for a player in a sequential game gives a list of choices, one for each decision node for that player.

Puny Poker (A+K+Q Game): Rules 2 players, each player gets 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.

Puny Poker (A+K+Q Game): Rules Betting: (“half street” game) Ante: 1 chip Player 1: bet 1, or check Player 2: call or fold † (, ) =

Puny Poker (A+K+Q Game): Rules Betting: (“half street” game) Ante: 1 chip Player 1: bet 1, or check Player 2: call or fold † (, ) = 2 if † = bet, ‡ = call, † > ‡ 1 if † = bet, ‡ = fold. 1 if † = check, † > ‡ . −1 if † = check, † < ‡ . −2 if † = bet, A‡ = call, † < ‡ .

Puny Poker (A+K+Q Game): Rules Betting: (“half street” game) Ante: 1 chip Player 1: bet 1, or check Player 2: call or fold ‡ , = −† (, )

Puny Poker: Game Rules 3 card deck: Ace > King > 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

Shuffle up and deal! 38

A+K+Q Analysis Better to be player 1 or player 2? Easy Decisions: Hard Decisions:

Game Payoffs Player 1: Ace King Queen Bet Check Bet Check Bet Check Player 2 Ace Call Fold King Call Fold Queen Call Fold

Game Payoffs (Player 1, Player 2) Player 1: Ace King Queen Bet Check Bet Check Bet Check Player 2 Ace Call (-2, +2) (-1,+1) (-2,+2) (-1,+1) Fold (+1,-1) (+1, -1) (+1,-1) (+1,-1) King Call (+2, -2) (+1, -1) (-2,+2) (-1,+1) Fold (+1, -1) (+1, -1) (+1,-1) (+1,-1) Queen Call (+2, -2) (+1, -1) (+2,-2) (+1,-1) Fold (+1, -1) (+1, -1) (+1,-1) (+1, -1)

Zero-Sum Game ∀ Š payoff. = 0 . ∈1

Player 1: Ace King Queen Bet Check Bet Check Bet Check Player 2 Ace Call -2 -1 -2 -1 Fold +1 +1 +1 +1 King Call +2 +1 -2 -1 Fold +1 +1 +1 +1 Queen Call +2 +1 +2 +1 Fold +1 +1 +1 +1 Payoffs for Player 1

Strategic Domination Strategy A dominates Strategy B if Strategy A always produces a better outcome than Strategy B regardless of the unknown state and other player’s action.

Player 1: Ace King Queen Bet Check Bet Check Bet Check Player 2 Ace Call -2 -1 -2 -1 Fold +1 +1 +1 +1 King Call +2 +1 -2 -1 Fold +1 +1 +1 +1 Queen Call +2 +1 +2 +1 Fold +1 +1 +1 +1 Eliminating Dominated Strategies

Player 1: Ace King Queen Bet Check Bet Check Player 2 Ace Call -1 -2 -1 King Call +2 -2 -1 Fold +1 +1 +1 Queen Fold +1 +1 Simplified (Player 1) Payoff Matrix

Player 1: Ace Queen Bet Bet Check Player 2 Ace Call -2 -1 King Call +2 -2 -1 Fold +1 +1 The Tough Decisions What if Player 1 never bluffs?

Expected Value

Expected Value . = Š Pr Ž payoff. () • ∈ •

Never Bluff Strategy Player 1: A K Q Bet Check Check Player 2 A Call -1 -1 K Fold/Call +1 -1 Q Fold +1 +1 † =

Never Bluff Strategy Player 1: A K Q Bet Check Check Player 2 A Call -1 -1 K Fold/Call +1 -1 Q Fold +1 +1 † = 1 3 1 + 1 3 − 1 2 + 1 2 + 1 3 −1 = 0

Player 1: Ace Queen Bet Bet Check Player 2 Ace Call -2 -1 King Call +2 -2 -1 Fold +1 +1 The Tough Decisions What if Player 1 always bluffs?

Always Bluff Strategy Player 1: A K Q Bet Check Bet Player 2 A Call -1 -2 K Call +2 -2 Fold +1 +1 +1 Q Fold +1 +1 †,“‡ ”•––— ˜™š› = 1 3 1 2 + 2 2 + 1 3 − 1 2 + 1 2 + 1 3 −2 = 0 †,“‡ •ž–Ÿ— ˜™š› = 1 3 1 2 + 1 2 + 1 3 − 1 2 + 1 2 + 1 3 −2 2 + 1 2 = − 1 6

Recap If player 1 never bluffs: If player 1 always bluffs: Looks like a break-even game for Player 1: is there a better strategy? † = 0, ‡ = 0 † = − 1 6 , ‡ = 1 6

Charge Project 3 is due tomorrow (Friday), 3:59pm 55