Credible continuations and commitment

Fund a refundable fake-chip bond, test an opponent’s backward response and separate binding incentives from an announcement.

18 minutes · advanced · Original Poker4Quant lesson

What you will learn

Solve a finite observed-move game backward, price an enforceable bond and distinguish sunk setup cost, refundable collateral and terminal receipts.

Prerequisites: Best responses · Information and incentives · Net payoff and sunk costs

Begin at the later choice

In this original perfect-information game, Row can Stay for receipts(2,5), or Explore. After Explore, Follower chooses Share for receipts(4,3)or Block for(0,1). Pairs list(Row,Follower). Without a commitment, Follower prefers Share because3>1. Row anticipates this and chooses Explore for4rather than Stay for2. The predicted Follower receipt is3.

A statement that Follower will Block does not change these later incentives. Check the action at the node where it would be chosen, even if an announced threat is supposed to keep play away from that node. This is backward reasoning about a supplied finite game with observed moves, not a behavioral prediction about actual people or a solution of full poker.

An enforced consequence can change the continuation

Before Row acts, Follower may escrow an integer bond Dbetween0and4. The known enforcement rule returns it after Stay or Block but forfeits it after Share. A nonzero bond also has a one-time setup cost c. With a bond, the later comparison becomes Share=3−Dversus Block=1, before the common setup cost. At D=2the choices tie; this exercise explicitly resolves the tie toward Share. At D≥3, Block is strictly better.

Row observes the bond and rule. If Share is predicted, Row chooses Explore for4rather than Stay for2. If Block is predicted, Row chooses Stay for2rather than Explore for0. An enforceable bond can therefore change the first action by changing the later incentives. It is the specified consequence, not merely the word commitment or earlier timing, that does the work.

Collateral is not the same as permanent cost

With setup cost1, D=3induces Stay. Follower receives5, recovers the bond and nets5−1=4relative to its starting wallet. D=4gives the same net but ties up more collateral, so the workshop’s stated secondary objective chooses3. A zero bond requires no setup and gives the no-bond baseline3. A refundable bond uses liquidity even when it is not finally lost.

With setup cost3, the deterring bonds net5−3=2, less than the no-bond3. The best policy is now D=0. Setup cost is sunk by the time Follower chooses Share/Block, so it must not distort that continuation comparison; it still matters for the earlier decision to set up a bond. The activity previews separate candidates rather than charging repeated fees to one accumulating wallet.

Keep hypothetical ledgers separate

Each candidate and terminal alternative starts with Row100,Follower100and a1000-chip treasury. The treasury collects setup/forfeited bond and pays terminal receipts. Total fake chips remain1200, with escrow separately visible until settlement. Stay, Share and Block are counterfactual endings, not three payouts to add together.

The exercise assumes perfect observability, enforceable forfeiture, the supplied tie rule and linear chip preferences. Removing enforcement, hiding the bond or changing payoffs creates a different game. There are no actual deposits, payments or real-money contests. The outcome audit grades reasoning and bookkeeping rather than realized profit.

Worked example

Setup cost is1. Fund D=3and compare the predicted Stay outcome with the counterfactual Share outcome.

Provisional funding: Row100,Follower96,escrow3,treasury1001. The total is1200.
Share would give Follower3−3=0before setup; Block gives1. Thus Block is the strict continuation preference.
Row anticipates Block and chooses Stay. Return bond3and pay receipts2to Row and5to Follower: Row102,Follower104,treasury994,escrow0.
The counterfactual Share branch forfeits bond3and pays receipts4and3: Row104,Follower99,treasury997,escrow0. Each branch conserves1200from its own identical baseline.

The predicted Follower net is4, improving on no-bond3by1. The forfeited bond belongs only to the Share counterfactual; do not subtract it from Stay.

Hands-on: Fund a bond and test backward responses

Test every finite collateral choice, predict the opponent’s continuation, choose the best least-collateral policy and reconcile three conserved alternative ledgers under two setup costs.

Reading commitment workshop…

Common mistakes

Believing an announcement changes later payoffs without enforcement.
Ignoring an unreached continuation when predicting the first move.
Treating D=2as a strict preference for Block despite the stated tie rule.
Subtracting refundable collateral as a permanent loss on every branch.
Ignoring setup cost in the initial policy comparison or counting it twice later.
Adding receipts from alternative worlds into one wallet.

Keep these ideas

Work backward from each later decision using the correct player’s incentives.
Binding consequences can change a continuation that cheap statements leave unchanged.
Separate a common sunk setup fee from a branch-specific forfeiture.
Compare net policy value, collateral use and the no-commitment baseline.
Finite toy-game conclusions require the stated observability, enforcement and tie assumptions.