Correct predictions are in blue. If we detect only a subset of a labelled sentence, we highlight the caught part as blue, the missing part light blue. False positives are in green and false negatives are in red.

Problem 108 (Winner Determination Problem) — Constraint detection

A set of bidders is denoted by N = -LCB- 1 , ... , n -RCB- , a set of items by M = -LCB- 1 , ... , m -RCB- . A bundle S is a set of items : $ S \ subset M$ . For a bundle S and a bidder i , we denote by vi -LRB- S -RRB- the package bid that bidder i makes for bundle S , i.e. , the maximal price that i announces to be willing to pay for S . An allocation of the items is described by variables $ xi -LRB- S -RRB- \ in -LCB- 0 , 1 -RCB- $ . The variable xi -LRB- S -RRB- is equal to one if and only if bidder i gets bundle S . An allocation $ -LRB- xi -LRB- S -RRB- | i \ in N , S \ subset M -RRB- $ is said to be feasible if it allocates no item more than once : $ \ sum _ -LCB- i \ in N -RCB- \ sum _ -LCB- S \ subset M , j \ in S -RCB- x_i -LRB- S -RRB- \ le 1 \ forall j \ in M$ , and at most one subset to every bidder : $ \ sum _ -LCB- S \ subset M -RCB- x_i -LRB- S -RRB- \ le 1 \ forall i \ in N$ . Given bids $ v_i , i = 1 , ... , n $ , the winner determination problem is the problem to compute $ x \ in argmax -LRB- \ sum _ -LCB- i \ in N -RCB- v_i -LRB- S -RRB- x_i -LRB- S -RRB- | x is a feasible allocation -RRB- $ .

Problem 108 (Winner Determination Problem) — Detection of the decisions and objects to be modeled

A set of bidders is denoted by N = -LCB- 1 , ... , n -RCB- , a set of items by M = -LCB- 1 , ... , m -RCB- . A bundle S is a set of items : $ S \ subset M$ . For a bundle S and a bidder i , we denote by vi -LRB- S -RRB- the package bid that bidder i makes for bundle S , i.e. , the maximal price that i announces to be willing to pay for S . An allocation of the items is described by variables $ xi -LRB- S -RRB- \ in -LCB- 0 , 1 -RCB- $ . The variable xi -LRB- S -RRB- is equal to one if and only if bidder i gets bundle S . An allocation $ -LRB- xi -LRB- S -RRB- | i \ in N , S \ subset M -RRB- $ is said to be feasible if it allocates no item more than once : $ \ sum _ -LCB- i \ in N -RCB- \ sum _ -LCB- S \ subset M , j \ in S -RCB- x_i -LRB- S -RRB- \ le 1 \ forall j \ in M$ , and at most one subset to every bidder : $ \ sum _ -LCB- S \ subset M -RCB- x_i -LRB- S -RRB- \ le 1 \ forall i \ in N$ . Given bids $ v_i , i = 1 , ... , n $ , the winner determination problem is the problem to compute $ x \ in argmax -LRB- \ sum _ -LCB- i \ in N -RCB- v_i -LRB- S -RRB- x_i -LRB- S -RRB- | x is a feasible allocation -RRB- $ .

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