Bridge and Mixed Strategy, an Example

I heard about this fascinating board from Spade14, one of the best bridge mentors I have ever had.

Problem

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In short, South reached a difficult 4♠ with 7 potential losers. After 6 rounds, they have a chance to minimize the damage or even make the contract.

The situation sounds obvious: either East or West has ♣A, which will be the 4th trick for them, beating the contract.

Here’s the catch: West wants more.

What are the common knowledge all players have at this point?

• South has the 2 remaining ♠;
• North has all the remaining ♥;
• If South has ♣A, the contract is making;
  • based on this, defenders need to assume South does not have ♣A;
  • That means South must have ♦AK, making 14 points in total, a reasonable inviting hand;
  • The location of ♣Q is unclear to defenders. But if South has it, they must be an unbalanced hand or they would have opened 1NT (15-17).

Let’s picture 4 different hands that West might be holding at this point (after 6 rounds):

• 1, With both ♣AQ, West has no choice but to play the ♣A, as playing any other card would make an entry for North, enabling the declarer to cash all the ♥;
• 2, With neither ♣A or ♣Q, West may easily play a small card. North can never cash the ♥ as East can overtake North’s ♣. The declarer will lose 2 ♠, 1 ♥, 2♦, 2 ♣: down 4, 400 points to East-West;
• 3, With ♣Q, West would still play a small card, leaving the guess to South. Had South played J, finessing the Q, East would take with the ♣A: down 1, 100 for East-West; Had South played K, guessing wrong, it would be down 4, 400 points for East-West;

In the first 3 scenarios, West has clear Dominant Strategy. Here comes the scenario No.4:

With ♣A in West and ♣Q in East

End Game

The remaining cards are:

West has two options:

• Play ♣A. This will guarantee down 1 as the result. (Now North has the entry to cash ♥)
• Take the risk and play a small ♣:
  • if declarer plays K, they will take all the remaining tricks, making 4♠, +620 for North-South
  • if declarer plays J, East can win with Q. West’s A can take over dummy’s K, forcing South to have 7 losers in total: +400 for East-West

To be safe, West should play ♣A to guarantee a positive score for their team. This, gives South an advantage: When West plays small, South will know that West does not have A, so South will definitely play ♣J to minimize the damage to -100.

However, knowing South will play J if West plays small, West can use it to break their consistent playing A and play a small, triggering South to play J and making East-West maximize their gain to +400.

Without dominant strategy in this scenario, here come bluffing and gambling.

Mixed Strategy

We can draw an Outcome Table in this scenario:

to be continued;