Volunteer's Dilemma

Nordstrom, Introduction to Game Theory · Section 4.3 · CC BY-SA 4.0

N players, everyone benefits if at least one person volunteers, but volunteering is costly. As N grows, each individual is less likely to act, and the probability that nobody volunteers increases. Also called "class-wide Chicken."

The payoff structure

Each player independently chooses: volunteer (cost = take 1 point) or don't (take 5 points). If at least one player volunteers, everyone gets their chosen amount. If nobody volunteers, everyone gets 0. Volunteering is individually costly but socially necessary.

Scheme

; Volunteer's Dilemma: N players choose to volunteer (1 pt) or not (5 pt)
; If at least one volunteers, all get their chosen amount
; If nobody volunteers, all get 0

(define (volunteer-payoffs choices)
  (let ((any-vol (memq 'vol choices)))
    (map (lambda (c)
           (cond
             ((not any-vol) 0)
             ((eq? c 'vol) 1)
             (else 5)))
         choices)))

; 3 players: one volunteers, two don't
(display "1 vol, 2 free-ride: ")
(display (volunteer-payoffs '(vol no no)))
(newline)

; Nobody volunteers
(display "nobody volunteers:  ")
(display (volunteer-payoffs '(no no no)))
(newline)

; All volunteer
(display "all volunteer:      ")
(display (volunteer-payoffs '(vol vol vol)))
(newline)

; The dilemma: free-riding gives 5 > 1 if someone else volunteers
; But if everyone reasons this way, nobody volunteers -> all get 0

The diffusion of responsibility

In a mixed-strategy Nash equilibrium, each player volunteers with some probability p. As N grows, each player's p shrinks: "surely someone else will do it." But the probability that nobody volunteers is (1-p) raised to the Nth power, which can increase even as N grows. More bystanders, less action.

Scheme

; As N grows, what happens to the probability nobody volunteers?
; In equilibrium, each player mixes. The indifference condition:
; payoff(vol) = payoff(no) => 1 = 5 * (1 - (1-p)^(N-1))
; Solving: (1-p)^(N-1) = 4/5
; So: p = 1 - (4/5)^(1/(N-1))

(define (equil-p n)
  (- 1.0 (expt 0.8 (/ 1.0 (- n 1)))))

(define (prob-nobody-volunteers n)
  (expt (- 1.0 (equil-p n)) n))

; Show how probability of no volunteer changes with N
(define (show-diffusion n)
  (if (<= n 10)
    (begin
      (display "N=")
      (display n)
      (display "  p(each)=")
      (display (number->string (equil-p n) 3))
      (display "  p(nobody)=")
      (display (number->string (prob-nobody-volunteers n) 3))
      (newline)
      (show-diffusion (+ n 1)))))

(show-diffusion 2)
; As N rises, individual p drops, but p(nobody) rises

Connection to Chicken

With N=2, the Volunteer's Dilemma reduces to Chicken. Volunteering is swerving: costly but safe. Not volunteering is going straight: you win big if someone else swerves, but if nobody does, everyone crashes. Nordstrom calls it "class-wide Chicken" for good reason.

Notation reference

Term	Scheme	Meaning
Volunteer	'vol	Accept cost (1 pt) to benefit group
Free-ride	'no	Take 5 pts if someone else volunteers
Diffusion	(expt (- 1 p) n)	Probability nobody acts, rises with N
Mixed equilibrium	(equil-p n)	Each player's indifference probability

Neighbors

Game theory foundations

🎲 Nordstrom 12 — Prisoner's Dilemma and Chicken (the 2-player cases)
🎲 Nordstrom 14 — Repeated Prisoner's Dilemma

Foundations (Wikipedia)

Translation notes

The equilibrium mixing calculation assumes symmetric players and the specific 1/5 payoff ratio from Nordstrom. Different payoffs shift the equilibrium probability but the qualitative result holds: more players, lower individual probability of volunteering, and the group failure probability can increase with group size.

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