Consider the following game.

Player B

Accept   Reject

  Accept 5, 7 13, X

Player A    Reject   7, 9 X, 6

What should be the minimum possible value of X for this game to be a Coordination Game?

(Note: if it must be higher than 10, enter 10)

Question

Consider the following game.

Player B

Accept      Reject

Accept     5, 7          13, X

Player A     Reject       7, 9          X,  6

What should be the minimum possible value of X for this game to be a Coordination Game?

(Note: if it must be higher than 10, enter 10)

BlackTom AI · Accepted Answer

First, let's lay out the payoff matrix described in the problem so we can analyze players' incentives clearly: 
- If A Accepts and B Accepts: (A payoff, B payoff) = (5, 7)
- If A Accepts and B Rejects: (13, X)
- If A Rejects and B Accepts: (7, 9)
- If A Rejects and B Rejects: (X, 6)

Next, we determine the best responses for each player given the other player's action, since a coordination game is typically defined by the existence of multiple pure-strategy Nash equilibria where players’ best responses align to support coordinated outcomes.

1) B's best response to A Accepting:
- If B Accepts: B gets 7
- If B Rejects: B gets X
So, when A Accepts, B prefers Accept if 7 ≥ X and prefers Reject if X > 7. At X = 7, B is indifferent (both actions yield 7).

2) B's best response to A Rejecting:
- If B Accepts: B gets 9
- If B Rejects: B gets 6
So, when A Rejects, B always prefers Accept (9 > 6).

3) A's best response to B Accepting:
- If A Accepts: A gets 5
- If A Rejects: A gets 7
So, when B Accepts, A prefers Reject (7 > 5).

4) A's best response to B Rejecting:
- If A Accepts: A gets 13
- If A Rejects: A gets X
So, when B Rejects, A prefers Accept if 13 ≥ X and prefers Reject if X > 13. At X = 13, A is indifferent.

With these best-response comparisons, we can look for potential pure-strategy Nash equilibria where each player's chosen action is a best response to the other's action:
- Candidate equilibrium (Accept, Accept): Not stable for A because if B Accepts, A would rather Reject (7 > 5).
- Candidate equilibrium (Reject, Reject): Not stable for B because if A Rejects, B would rather Accept (9 > 6).
- Candidate equilibrium (Accept, Reject): For this to be an equilibrium, A must not want to switch given B Rejects, and B must not want to switch given A Accepts. From above, A would prefer Accept when X ≤ 13 and would prefer Reject only if X > 13. Since B Rejects, the condition for A to stay with Accept is X ≤ 13. For B to stay with Reject given A Accepts, B must have 7 ≥ X. Combining these: X ≤ 7 ensures B’s best response includes Reject, and X ≤ 13 ensures A’s best response includes Accept. The tightest requirement is X ≤ 7, with equality allowing indifference for B at X = 7.
- Candidate equilibrium (Reject, Accept): For this to be an equilibrium, A must not want to switch given B Accepts, and B must not want to switch given A Rejects. From above, given B Accepts, A prefers Reject (7 > 5), so this is satisfied. Given A Rejects, B prefers Accept (9 > 6), so this is also satisfied regardless of X. Therefore (Reject, Accept) is always a Nash equilibrium for any X.

From the above analysis, we see that to have the two coordination-type equilibria (Reject, Accept) and (Accept, Reject) both exist as best responses (i.e., a coordination game with two clear coordination points), the key constraint comes from the (Accept, Reject) path. For (Accept, Reject) to be an equilibrium, we need B not to deviate when A Accepts, i.e., X ≤ 7 would keep B from preferring Reject; but since at X = 7 B is indifferent (7 equals the payoff from Accept), the equilibrium still holds. Pushing X any higher than 7 would still allow (Reject, Accept) as a Nash equilibrium, and (Accept, Reject) would remain an equilibrium as long as A is willing to Accept when B Rejects (which requires X ≤ 13). The combination that ensures both coordination points exist with the smallest X is X = 7, where indifference allows both coordination outcomes to be supported as equilibria.

Therefore, the minimum value of X that makes the game a coordination game (in the sense of supporting the two coordination equilibria) is 7.

COMM_V 295 105 106 2025W1 Participation/Math Quiz #2

Consider the following game. Player B Accept Reject Accept 5, 7 13, X Player A Reject 7, 9 X, 6 What should be the minimum possible value of X for this game to be a Coordination Game? (Note: if it must be higher than 10, enter 10)

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