# Steering User Behavior with Badges

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13 minimize x subject to f( x) = p x 2 2 θ C T x + λ(1 T x 1 + x n+1 ) x j 0, j = 1, 2,..., n Now, we can expand the norm to express f( x) as f( x) = p T p 2 p T x + x T x θc T x + λ(1 T x (1 x n+1 )) By taking partial derivatives and setting them to zero, we can express the solution in terms of λ: δf( x) δx i = 2 p i + 2 x i θc i + λ = 0 Now we have expressed x in terms of λ, so we can simply substitute back into our original function: θc T x x p 2 2 (x n+1 p n+1 ) 2 1 θx n+1 = = ( pt p z λ 2 + λ 2 n λ /4) (1 1 T z λ + λn λ /2 p n+1 ) 2 1 θ(1 1 T z λ + λn λ /2) We have now transformed the problem to an equivalent onedimensional optimization problem. The final expression is a piecewise quadratic-over-linear function of λ with breakpoints at 2z i for i = 1, 2,..., n. We solve each piece with elementary calculus and take the max over all pieces, which is then the global optimum of our original problem. x i = 1 2 [2 pi + θc i λ] + where [x] + = max(x, 0) enforces the non-negativity constraint. It will be convenient to define the vector z such that z i = p i + θc i /2. For { any vector v, let v λ denote the vector such that vλ i v i 2z i > λ =. Then we can rewrite 0 otherwise p x 2 2 θc T x as: p x 2 2 θc T x = = p T p ( p i (2z i λ)) i:2z i >λ = p T p p T (2 p λ + θc λ λ1 λ ) λ 2 i:2z i >λ i:2z i >λ (2z i λ) 2 C i (2z i λ) p λ + θc λ λ1 λ 2 2 θ 2 CT (2 p λ + λc λ λ1 λ ) where 1 = [ ] T is the vector of all 1s. Now we can expand the norm and use the fact that for any vectors a and b, a T b λ = b T a λ = b T λ a λ to simplify further: p x 2 2 θc T x = = p T p 1 2 pt (2 p λ + θc λ λ1 λ ) θ 4 CT (2 p λ + θc λ λ1 λ ) λ 4 1T (2 p λ + θc λ λ1 λ ) = p T p p T p λ θ p T C λ θ2 4 CT C λ + λ2 4 n λ = p T p p λ + θ 2 C λ λ2 4 n λ = p T p z λ λ2 4 n λ where n λ = {i : 2p i + θc i > λ}. Finally, we derive x n+1 using the fact that x n+1 = 1 1 T x: x n+1 = 1 1 T x = 1 1 (2 p i + θc i λ) 2 i:2z i >λ = 1 z i + n λ 2 λ i:2z i >λ = 1 1 T z λ + n λ 2 λ

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