From attention to purchase

Marketers have drawn the path from attention to purchase as a funnel for more than a century. One early version, attributed to the advertising pioneer E. St. Elmo Lewis, ran Attention, Interest, Desire, Action (Strong 1925). On a short-video or live-commerce platform the stages can be measured: an impression (the video was shown), a watch (the viewer stayed past the first seconds), engagement (the viewer watched most of it, or commented), a visit to the shop or profile, and an order.

A funnel drawn as a straight line hides two facts. Most viewers leave at every stage. And some go back: a shop visitor returns to watch another video before deciding, an engaged viewer scrolls on to another clip from the same creator. A Markov chain captures both.

The model

Each viewer is in one of a few states. At every step they move to another state with fixed probabilities that depend only on the state they are in. Two states, Buy and Exit, are absorbing: once there, the viewer stays. The others are transient. Collect the transition probabilities between transient states in a matrix $Q$, and the probabilities of moving from each transient state to Buy in a vector $r$. The probabilities of eventually buying, starting from each state, are the solution $x$ of

$$(I – Q)\,x = r ,$$

and the matrix $N = (I – Q)^{-1}$, the fundamental matrix, gives the expected number of visits to each state before absorption (Kemeny and Snell 1960).

# Absorbing Markov chain of one viewer: Impression -> Watch (3 s) -> Engage (half the video) ->
# Shop visit.
# Every state can also end in Exit; Shop visit can end in Buy. Numbers are illustrative, not
# measured.
S = ["Impression", "Watch", "Engage", "Shop"]
Q = [[0, .30, 0, 0],        # transitions between the four transient states (row -> column)
     [0, 0, .35, 0],
     [0, .10, 0, .20],      # an engaged viewer may watch another video, or open the shop
     [0, 0, .25, 0]]        # a shop visitor may go back to the content
buy = [0, 0, 0, .15]        # Shop -> Buy; whatever is left in each row is Exit

def solve(A, b):            # Gaussian elimination, small dense systems
    n = len(A); M = [row[:] + [v] for row, v in zip(A, b)]
    for c in range(n):
        p = max(range(c, n), key=lambda r: abs(M[r][c])); M[c], M[p] = M[p], M[c]
        for r in range(n):
            if r != c:
                f = M[r][c] / M[c][c]
                M[r] = [x - f * y for x, y in zip(M[r], M[c])]
    return [M[i][n] / M[i][i] for i in range(n)]

def p_buy(Q, buy):          # absorption probability: (I - Q) x = buy
    IQ = [[(i == j) - Q[i][j] for j in range(4)] for i in range(4)]
    return solve(IQ, buy)

def steps(Q):               # expected number of steps before absorption: (I - Q) t = 1
    IQ = [[(i == j) - Q[i][j] for j in range(4)] for i in range(4)]
    return solve(IQ, [1, 1, 1, 1])

pb, st = p_buy(Q, buy), steps(Q)
for s, x, t in zip(S, pb, st):
    print(f"from {s:10s} P(buy) = {x:7.4f}   expected steps before leaving or buying = {t:4.2f}")
print(f"orders per 10,000 impressions: {10000 * pb[0]:.0f}")

# Which link is worth improving? Raise one forward probability by 10% of itself (taken from
# Exit).
print("\nlink improved by 10%       orders per 10,000   change")
base = p_buy(Q, buy)[0]
links = [("Impression -> Watch", 0, 1), ("Watch -> Engage", 1, 2), ("Engage -> Shop", 2, 3), ("Shop -> Buy", 3, None)]
for name, i, j in links:
    Q2 = [row[:] for row in Q]; b2 = buy[:]
    if j is None: b2[i] *= 1.1
    else:         Q2[i][j] *= 1.1
    new = p_buy(Q2, b2)[0]
    print(f"{name:24s} {10000 * new:12.1f} {new / base - 1:12.1%}")
# from Impression P(buy) =  0.0034   expected steps before leaving or buying = 1.45
# from Watch      P(buy) =  0.0115   expected steps before leaving or buying = 1.50
# from Engage     P(buy) =  0.0328   expected steps before leaving or buying = 1.42
# from Shop       P(buy) =  0.1582   expected steps before leaving or buying = 1.36
# orders per 10,000 impressions: 34
#
# link improved by 10%       orders per 10,000   change
# Impression -> Watch              37.9        10.0%
# Watch -> Engage                  38.0        10.4%
# Engage -> Shop                   38.1        10.6%
# Shop -> Buy                      37.9        10.0%

With these numbers, 10,000 impressions produce about 34 orders. Two results deserve attention.

Every link matters about equally. Raising any single link by 10 per cent raises orders by about 10 per cent. In a straight chain, the probability of reaching the end is the product of the links, so each link has an elasticity of exactly one; there is no single “magic” stage. The loops make the middle links slightly more valuable, because an improvement there is used more than once.

Loops are trust. The return paths, from the shop back to the content and from one video to the next, are the model’s representation of a viewer gathering evidence before buying. Removing them lowers conversion even though no forward link changes. The value of a body of content, rather than a single video, shows up here.

Where dopamine comes in, and where it does not

It is common to describe feeds as “dopamine machines”. The neuroscience behind the phrase is real but narrower than the slogan.

The relevance to a funnel is indirect. Content that keeps delivering small unexpected rewards, a surprising cut, a reveal, a price drop, keeps viewers in the “watch” state longer and raises the probability of moving forward, much as variable-ratio schedules sustain behaviour in the laboratory. But wanting is not trust. A viewer can be held by a feed and still leave at the shop page because nothing has given them reason to believe the product is good.

Time in a state: dwell and decay

The simple chain treats each step as one decision. In practice, how long a viewer stays matters too. If the chance of leaving a live stream in any given minute is a constant $h$, the time a viewer stays follows an exponential distribution with mean $1/h$: a stream that cuts the per-minute leaving rate from 10 to 8 per cent raises the average stay from 10 to 12.5 minutes, a quarter more exposure at the same audience size. Dwell time is valuable not for itself but because it multiplies the chances to move forward in the chain.