Graph playground
\(1\)
\(n\)
\(\vdots\)
\(v_1 \sim F_1\)
\(v_n \sim F_n\)
\(b_1(\rm v_1)\)
\(b_n(\rm v_n)\)
\(\vdots\)
agents
strategy profile
value distribution
\(\bm F\)
\(\rm b_1\)
\(\rm b_{\it n}\)
actions
i.e. bids
allocation
\({\bf x}=(\rm x_1, ..., x_n)\)
payment
\({\bf p}=(\rm p_1, ..., p_n)\)
\(\vdots\)
\(\bf b\)
\(\bf b\)
or: \({\bm b(\bm v) } = \bf b\)
mechanism
allocation is a function of the bids
\(\bf x(b) = x(\bm b(v))\)
payment is a function of the bids
\(\bf p(b) = p(\bm b(v))\)
ex ante
ex post
interim
three stages of a Bayesian game
agents know distribution of types
[common prior]
\(v_i \sim\)\( F_i\)
agents learn their types
\({\rm v}_i\)
game not played
game played;
types and actions all known
also outcome and payments
\(\rm v\)
\(\rm x\)
\(\rm v_i\)
\(\rm v_j\)
\(\rm x_i\)
\(\rm x_j\)
\(\rm v_j (x_i - x_j)\)
\(0\)
\(q\)
\(x\)
\(1\)
\(s_L\)
\(s_H\)
\(b\)
quality: skill times budget
allocation: probability that one is going to get the item
\(x \in [0, 1]\)
\(q = s \cdot b\)
\(q\)
\(x\)
\(1\)
\(s_L\)
\(s_H\)
Two skills; budget \(b=1\)
\(q\)
\(x\)
\(1\)
\(s_L\)
\(s_H\)
Feasible region
constraints agents face:
- non-negative utility
- does not exceed budget
WTH is this feasible?
\(q\)
\(x\)
\(1\)
\(s_L\)
\(s_H\)
Two types of budget \(b_L, b_H\)
\(b_L < 1\) by def
\(s_Lb_L\)
\(q\)
\(x\)
\(1\)
\(s_L\)
\(s_H\)
Feasible region now
\(b_L < 1\) by def
\(s_Lb_L\)
\(q\)
\(x\)
\(1\)
\(s_L\)
\(s_H\)
\(s_Lb_L\)
\(s_Hb_L\)
optimal menu 1
\(\{(s_Lb_L + \epsilon, 1)\}\)
\(q\)
\(x\)
\(1\)
\(s_L\)
\(s_H\)
optimal menu 2
\(\{(s_Hb_L, s_Hb_L/s_L - \epsilon), (s_L\min\{b_H, 1\}+\epsilon, 1)\}\)
\(s_Lb_L\)
\(s_Hb_L\)
When \( b_H < 1\)
\(q\)
\(x\)
\(1\)
\(s_L\)
\(s_H\)
optimal menu 2
\(\{(s_Hb_L, s_Hb_L/s_L - \epsilon), (s_L\min\{b_H, 1\}+\epsilon, 1)\}\)
\(s_Lb_L\)
\(s_Hb_L\)
When \( b_H \geq 1\)
\(q\)
\(x\)
\(1\)
\(s_L\)
\(s_H\)
Generalizing...
\(s_Lb_L\)
\(s_Hb_L\)
i.e. let \(b_L\) take different values
\(q\)
\(x\)
\(1\)
\(s_L\)
\(s_H\)
optimal menu 2
\(\{(s_Hb_L, s_Hb_L/s_L - \epsilon), (s_L\min\{b_H, 1\}+\epsilon, 1)\}\)
\(s_Lb_L\)
\(s_Hb_L\)
depending on the value of \(b_H\)
\(q\)
\(x\)
\(1\)
\(s_L\)
\(s_H\)
\(b\)
\(q\)
\(x\)
\(1\)
\(s_L\)
\(s_H\)
\(b\)
\(q\)
\(x\)
\(1\)
\(s_L\)
\(s_H\)
Generalizing...
\(s_Lb_L\)
\(s_Hb_L\)
how do we justify this jump???
jump
first type of jump
\(q\)
\(x\)
\(1\)
\(s_L\)
\(s_H\)
Jump 2
\(s_Lb_i\)
\(s_Hb_j\)
\(q\)
\(x\)
\(1\)
\(s_L\)
\(s_H\)
Jump 3
\(s_Lb_i\)
\(s_Hb_j\)
Graph playground
By Sheng Long
Graph playground
- 18
