深夜數學題9

咁就要睇probability

只要a不是50 b都會弄它

答案係 A 首叫 $41 既前提係, A 同 B 都遵從以下:

1. 首要目的係最大化已利益
2. 係自己利益相同下, 最少化對手利益

而且假設雙方都係理性 (即係遵從上面兩條)

另外, $250 budget 係關鍵, 如果唔係 $250 會唔同哂

如果 budget 係 $100, 咁一定係搶住叫 $91 (明顯)

如果 budget 係 $110, 會突然唔同哂: A 可以大膽首叫 $10, 首先 B 唔可能叫 > $100 (由無賺無蝕變蝕至少 $1); 然後 B 如果叫 [$20, $100] 咁 A 叫 $110 B 即刻收皮 (A 唔會蝕多過 $10)

陣間繼續寫

p = probability of opponent dropping out

E(r(i))= p(i)* $100 + (1-p(i)) * $0 - E(r(i-1))

你唔諗a出完101 b咪一定同你跟落去 咁你a9咪唔成立lo

a91一樣 b一定叫150 因為佢知你a會再叫 所以a91都唔成立

你地簡單一句loop就算
無諗過點loop法 最後邊個得益
倒推結果 a唔會咁做害自己

呢題dollar auction 最基本要用到backward thinking ，如果唔知咩係backward thinking 可以參考一下以下pirate game

There are 5 rational pirates (in strict order of seniority A, B, C, D and E) who found 100 gold coins. They must decide how to distribute them.

The pirate world's rules of distribution say that the most senior pirate first proposes a plan of distribution. The pirates, including the proposer, then vote on whether to accept this distribution. If the majority accepts the plan, the coins are dispersed and the game ends. In case of a tie vote, the proposer has the casting vote. If the majority rejects the plan, the proposer is thrown overboard from the pirate ship and dies, and the next most senior pirate makes a new proposal to begin the system again. The process repeats until a plan is accepted or if there is one pirate left.[1]

Pirates base their decisions on four factors. First of all, each pirate wants to survive. Second, given survival, each pirate wants to maximize the number of gold coins each receives. Third, each pirate would prefer to throw another overboard, if all other results would otherwise be equal.[2] And finally, the pirates do not trust each other, and will neither make nor honor any promises between pirates apart from a proposed distribution plan that gives a whole number of gold coins to each pirate.

點解b要弄a先？a有咩好處先？如果大家想弄對方又有咩條件支持先？假設b lose $1 = a gain $x 或者b 係fully rational 要用probability 做呢題

b唔係fully rational 就要用probability*

假設2個人都係絕對理性
A出10蚊就得
B係呢個情況冇可能會投標，
因為A出左10蚊之後，下一次標價係110內
因為B目標係賺錢， ie，標價係100內
佢要確保自己標價係100內，同時A唔還標先有錢賺
但呢個係冇可能的
只要B會價100蚊內，A絕對會還標出110確保自己有機會賺錢同時虧少過10蚊
而係A出110後，B再還標出就會開始虧錢
（因為假設佢地係絕對理性，所以B係唔會還第一次標）
所以係10蚊
1-9蚊會因為B出價100蚊而唔還標，（一還就會虧起碼10蚊）虧硬

肯撚定岩 太簡單

你諗得太簡單

b要弄a因為佢知a會跟上來 而最後a會比b吃虧

a係絕對會跟你 因為可以減少吃虧

你可以諗下有negative payoff

可以試玩: 我(B) 出 $41

不如咁諗，A要叫咩先會令B叫$250?

咪講左絕對理性囉，姐係唔會做negative payoff呀

重覆哩個process 到搵到答案 （$250--> $0)

大於150 then?

A出$10 嗰一刻就有$10 sunk cost

咁A一定會嗌110以內，因為唔會虧多過10蚊
B再還價就一定會虧錢，就算嗌120都虧20蚊
因為絕對理性，foresee到會虧錢，咁B就唔會嗌第一次

同埋我想講 250最後a要話事 唔會比b 投到250 b一係唔投 一投250係坐和望炒a 所以250是屬於a

岩呀，所以下次就會嗌110以內，which結果冇賺冇虧
如果B第一次有點標，佢最尾都一定要用多過110蚊去標贏，虧硬，所以佢第一次都唔會回標