PS1sol

The Saga of Big Monkey (BM) and Little Monkey (LM)

  • setup

    • Warifruit is worth 1010 kilocalories (Kc).

    • BM’s cost to climb is 22 Kc; LM’s climbing cost is negligible.

    • If both climb, BM gets most of the fruit: BM = 77 Kc, LM = 33 Kc.

    • If only LM climbs, after LM returns BM has eaten 99 Kc, LM gets 11 Kc.

    • If BM climbs, LM can consume about 44 Kc before BM climbs down and chases him away, leaving BM with 66 Kc.

    • Actions: W = wait (don’t climb), C = climb.

  • a) Extensive form when BM decides first

    • BM at the root chooses between WW and CC.

    • If BM = WW, LM faces a decision node with actions WW or CC.

    • If BM = CC, LM faces a second decision node with actions WW or CC.

    • Terminal payoffs (BM, LM):

    • BM = WW, LM = WW: (0,0)(0,0)

    • BM = WW, LM = CC: (9,1)(9,1)

    • BM = CC, LM = WW: (4,4)(4,4)

    • BM = CC, LM = CC: (5,3)(5,3)

  • b) Strategies for each monkey

    • BM: ext{BM} o \{W, C}

    • LM: LM’s contingent plans across LM’s two decision nodes → WW,WC,CW,CC{WW, WC, CW, CC}

    • WW = wait after BM = W; wait after BM = C

    • WC = wait after BM = W; climb after BM = C

    • CW = climb after BM = W; wait after BM = C

    • CC = climb after BM = W; climb after BM = C

  • c) Normal-form (strategic form) representation

    • BM has 2 strategies: W,CW, C

    • LM has 4 strategies: WW,WC,CW,CCWW, WC, CW, CC

    • Payoff matrix (BM strategies as rows, LM strategies as columns):

    • BM = WW, LM = WWWW: (0,0)(0,0)

    • BM = WW, LM = WCWC: (0,0)(0,0)

    • BM = WW, LM = CWCW: (9,1)(9,1)

    • BM = WW, LM = CCCC: (9,1)(9,1)

    • BM = CC, LM = WWWW: (4,4)(4,4)

    • BM = CC, LM = WCWC: (5,3)(5,3)

    • BM = CC, LM = CWCW: (4,4)(4,4)

    • BM = CC, LM = CCCC: (5,3)(5,3)

  • d) How many Nash equilibria (NE) in this BM-first game? What are they?

    • There are 3 NE:

    • (W, CW)

    • (W, CC)

    • (C, WW)

    • In terms of strategies:

    • NE 1: BM = WW, LM = CWCW

    • NE 2: BM = WW, LM = CCCC

    • NE 3: BM = CC, LM = WWWW

  • e) Assume LM decides first. Redo a–d

    • Extensive form when LM decides first

    • LM’s choice at the root: WW or CC; BM has contingent decisions after LM’s move across two BM decision nodes.

    • Strategies: LM ∈ \{W, C\}; BM ∈ \{WW, WC, CW, CC\}.

    • Payoff matrix (LM strategies as rows, BM strategies as columns):

    • If LM = WW, BM = WWWW: (0,0)(0,0)

    • LM = WW, BM = WCWC: (0,0)(0,0)

    • LM = WW, BM = CWCW: (4,4)(4,4)

    • LM = WW, BM = CCCC: (4,4)(4,4)

    • If LM = CC, BM = WWWW: (1,9)(1,9)

    • LM = CC, BM = WCWC: (3,5)(3,5)

    • LM = CC, BM = CWCW: (1,9)(1,9)

    • LM = CC, BM = CCCC: (3,5)(3,5)

    • NE (LM-first): 3 equilibria

    • (W, CW)

    • (W, CC)

    • (C, WW)

  • f) Assume the monkeys decide simultaneously. Redo a–d

    • Simultaneous-decision game with LM ∈ {W, C} and BM ∈ {W, C}.

    • Payoff matrix (LM rows, BM columns):

    • LM = W, BM = W: $(0,0)$

    • LM = W, BM = C: $(4,4)$

    • LM = C, BM = W: $(1,9)$

    • LM = C, BM = C: $(3,5)$

    • NE in this simultaneous game: two equilibria

    • (W, C)

    • (C, W)

  • g) Why not redo e when solving f?

    • The reason is that the extensive-form representation shown already captures the contingent move structure and the corresponding normal-form (strategic) representation is the same whether you derive it from the BM-first or the LM-first extensive form. The simultaneous-move game has a single normal-form representation that yields the same NE set, so redrawing the extensive form assuming the other player moves first would duplicate effort without changing the strategic (normal-form) outcomes.

  • Note on terminology used in this section: W = wait or don’t climb, C = climb; LM = Little Monkey; BM = Big Monkey.

Erste / Zweite (First and Second) game

  • a) Strategies

    • Erste (First mover): They have 3 decision nodes; thus a strategy specifies a choice at the first node and choices for the second two nodes. Strategies listed:

    • AGG, AGH, AHG, AHH, BGG, BGH, BHG, BHH

    • Zweite (Second mover): They have two decision nodes; thus a strategy lists a planned choice for each node. Strategies:

    • ce, cf, de, df

  • b) Normal-form representation

    • Strategy sets:

    • Erste: {AGG, AGH, AHG, AHH, BGG, BGH, BHG, BHH}

    • Zweite: {ce, cf, de, df}

    • Payoff table (Erste strategies as rows, Zweite strategies as columns). Example entries:

    • Erste AGG, Zweite ce: $(6,1)$

    • Erste AGG, Zweite cf: $(6,10)$

    • Erste AGG, Zweite de: $(9,3)$

    • Erste AGG, Zweite df: $(9,3)$

    • Erste AGH, Zweite ce: $(6,1)$

    • Erste AGH, Zweite cf: $(6,1)$

    • Erste AGH, Zweite de: $(2,5)$

    • Erste AGH, Zweite df: $(2,5)$

    • Erste AHG, Zweite ce: $(7,9)$

    • Erste AHG, Zweite cf: $(7,9)$

    • Erste AHG, Zweite de: $(9,3)$

    • Erste AHG, Zweite df: $(9,3)$

    • Erste AHH, Zweite ce: $(7,9)$

    • Erste AHH, Zweite cf: $(7,9)$

    • Erste AHH, Zweite de: $(2,5)$

    • Erste AHH, Zweite df: $(2,5)$

    • Erste BGG, Zweite ce: $(8,3)$