Why Players Randomize
A player who chooses the same action every time can become predictable. In some strategic situations, choosing among actions according to a probability distribution prevents an opponent from taking advantage of a fixed choice. A mixed strategy describes that randomization; it does not mean that the player is uncertain about the payoffs or changes preferences from one play to the next.
The central condition is indifference. If a player deliberately assigns positive probability to two actions, those actions must give the player the same expected payoff against the other players’ strategies. Otherwise, the player could improve by putting more probability on the action with the higher expected payoff. Indifference alone is not a complete equilibrium test: an action assigned zero probability must not offer a still higher payoff.
For example, suppose a firm can choose either a standard delivery plan or an expedited plan. A mixed strategy might assign probability 0.3 to standard delivery and 0.7 to expedited delivery. In a one-shot simultaneous game, assume each player’s randomization follows the announced probabilities, players maximize expected payoff, and their randomizations are independent unless the model explicitly says otherwise. Expected payoff is the probability-weighted average of the payoffs for the possible outcomes.
A strategy describes a probability distribution over actions, not a probability distribution over payoff values. Once actions are selected, the game produces an outcome and the players receive the payoffs associated with it. The expected payoff is calculated before those random choices are realized.
Expected Payoffs Against a Randomizing Opponent
Use the payoff matrix to read a player’s payoff for each action combination. If the opponent chooses action \(j\) with probability \(q_j\), the expected payoff from one of your actions is the sum of your payoffs against each opponent action, weighted by the corresponding \(q_j\). The probabilities must refer to the opponent’s actions, not to the payoffs.
Here \(a\) is one of the player’s actions, \(j\) indexes the opponent’s possible actions, \(q_j\) is the probability of action \(j\), and \(u(a,j)\) is the player’s payoff if the action pair \((a,j)\) occurs. To evaluate a mixed strategy, calculate this expected payoff for each action in its support and for any unused action that might provide a profitable deviation.
Worked Example: Choosing a Best Response to a Mixed Strategy
A supplier chooses either Rapid or Thorough processing. A buyer chooses either Routine or Urgent service. The supplier’s payoffs are shown below. The buyer independently chooses Routine with probability 0.4 and Urgent with probability 0.6. Which supplier action has the higher expected payoff, and can the supplier mix between the two actions as a best response?
| Supplier action | Routine (0.4) | Urgent (0.6) |
|---|---|---|
| Rapid | 8 | 2 |
| Thorough | 5 | 6 |
Method. Compute the supplier’s expected payoff from each pure action using the buyer’s stated probabilities, then compare them. This avoids treating the two buyer actions as equally likely just because the table has two columns.
For Rapid, the expected payoff is \(0.4(8)+0.6(2)=3.2+1.2=4.4\). For Thorough, it is \(0.4(5)+0.6(6)=2.0+3.6=5.6\).
Thorough gives the higher expected payoff: \(5.6>4.4\). Therefore, the supplier’s unique best response among these two actions is Thorough. A strategy assigning positive probability to Rapid as well as Thorough is not a best response, because shifting that probability from Rapid to Thorough raises expected payoff. The supplier should not mix between them in response to this buyer strategy.
Check. The probabilities sum to \(0.4+0.6=1\). The expected payoff difference is \(5.6-4.4=1.2\), in favor of Thorough. The common mistake avoided is using an unweighted average, which would incorrectly ignore the buyer’s unequal probabilities.
Indifference in an Equilibrium Mix
In a Nash equilibrium, each player’s strategy is a best response to the other players’ strategies. This gives a precise reason for indifference among actions in a player’s support. Suppose two actions both receive positive probability, but one has a strictly higher expected payoff against the opponents’ strategies. The player could increase the probability of the higher-payoff action and reduce the probability of the lower-payoff action. Since expected payoff is a probability-weighted average, this change would raise the player’s expected payoff. The original mix would not be a best response.
The argument follows directly from weighted averages. If a player puts positive probability on actions \(a\) and \(b\), and \(EU(a)>EU(b)\), then moving a positive amount of probability from \(b\) to \(a\) increases expected payoff by that amount multiplied by \(EU(a)-EU(b)\). A player choosing a best response cannot have such a profitable change. The same reasoning applies if several actions receive positive probability: each must attain the maximum expected payoff available to that player.
This condition does not say that every available action must yield the same expected payoff. An action assigned probability zero may yield less than the common support payoff. It may also tie that payoff, in which case adding it to the support can remain a best response. But an unused action that yields more would be a profitable unilateral deviation, contradicting equilibrium.
Worked Example: Identifying the Support of a Best Response
A retailer chooses either Launch, Wait, or Delay. Its distribution partner chooses among three contract conditions. The partner’s probabilities and the retailer’s payoffs are given below. Determine which retailer actions can receive positive probability in a best response.
| Retailer action | Basic (0.25) | Flexible (0.50) | Restricted (0.25) |
|---|---|---|---|
| Launch | 10 | 4 | 2 |
| Wait | 6 | 6 | 2 |
| Delay | 8 | 3 | 0 |
Method. Calculate expected payoff for all three actions. A best-response mix can include only actions that reach the highest expected payoff; this comparison checks the unused action as well as the proposed support.
For Launch, the expected payoff is \(0.25(10)+0.50(4)+0.25(2)=2.50+2.00+0.50=5.00\). For Wait, it is \(0.25(6)+0.50(6)+0.25(2)=1.50+3.00+0.50=5.00\). For Delay, it is \(0.25(8)+0.50(3)+0.25(0)=2.00+1.50+0=3.50\).
Launch and Wait tie at \(5.00\), while Delay yields \(3.50\). The retailer may choose Launch alone, Wait alone, or any probability distribution over Launch and Wait as a best response. It cannot assign positive probability to Delay in a best-response mix: moving that probability to either Launch or Wait raises expected payoff. This identifies a best response to the partner’s strategy, not by itself a full equilibrium, since the partner’s strategy must also be a best response to the retailer’s.
Check. The partner’s probabilities sum to \(0.25+0.50+0.25=1\), and the highest calculated payoff is \(5.00\). The common mistake avoided is checking equality only among the actions one hoped to include, without testing whether an omitted action pays more.
Checking a Proposed Mixed-Strategy Profile
To check a proposed profile in a two-player game, calculate each player’s expected payoff from every available pure action, holding the opponent’s strategy fixed. The actions each player uses with positive probability must tie at the maximum. If a player has only two available actions and both are in the support, both must attain that same maximum. When this test holds for both players, each is best responding to the other.
Worked Example: Testing a Candidate Pair of Mixed Strategies
Two firms simultaneously choose one of two service designs. The first firm’s payoff is listed first in each cell, and the second firm’s payoff is listed second. A proposed strategy pair has the first firm choose North with probability 0.25 and South with probability 0.75; the second firm chooses Left and Right with probability 0.50 each. Check whether each firm is best responding.
| First firm | Left | Right |
|---|---|---|
| North | (8, 3) | (2, 9) |
| South | (4, 7) | (6, 5) |
Method. For the first firm, calculate the payoff to North and South against the second firm’s 50–50 strategy. For the second firm, calculate the payoff to Left and Right against the first firm’s 25–75 strategy. This checks every pure deviation for both players.
The first firm’s expected payoff from North is \(0.50(8)+0.50(2)=4+1=5\). From South it is \(0.50(4)+0.50(6)=2+3=5\). Thus both actions used by the first firm yield 5.
The second firm’s expected payoff from Left is \(0.25(3)+0.75(7)=0.75+5.25=6\). From Right it is \(0.25(9)+0.75(5)=2.25+3.75=6\). Thus both actions used by the second firm yield 6.
Each player’s available actions tie at that player’s maximum expected payoff. Neither player can gain by switching to the other pure action, so the proposed pair is a Nash equilibrium. The expected payoffs at the profile are 5 for the first firm and 6 for the second. For example, the first firm’s payoff averaged across outcomes is \(0.25[0.50(8)+0.50(2)]+0.75[0.50(4)+0.50(6)]=0.25(5)+0.75(5)=5\).
Check. Each player’s probabilities are nonnegative and sum to 1; both players’ pure-action payoffs tie as calculated. The common mistake avoided is checking indifference for only one player and then declaring a two-player strategy profile an equilibrium.
Common Mistakes and Why the Condition Matters
Indifference is useful because it turns a claim about a randomizing player’s incentives into a calculation that can be checked. If a proposed strategy assigns positive probability to actions with unequal expected payoffs, it cannot be a best response. If an unused action has a larger expected payoff than all actions in the support, the strategy also fails the best-response test. The condition applies to expected payoffs given the opponents’ strategies, not to the payoffs in individual outcome cells.
- Using an unweighted average. Average each action’s payoffs using the opponent’s stated probabilities. Equal weighting is correct only when those probabilities are equal.
- Assuming every action must tie. Indifference is required among actions in the support. An unused action may have a lower payoff; it must not have a higher one.
- Confusing a best response with an equilibrium. A player’s mix can be a best response to a given opponent strategy even when that opponent strategy is not itself a best response.
- Reading an outcome payoff as an expected payoff. A payoff in one cell applies only when that action combination occurs. A mixed strategy requires weighting the relevant cell payoffs by outcome probabilities.
These ideas are aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus. In a game-theory problem, first identify whose payoff is being evaluated, then condition on the other players’ strategies, calculate expected payoffs, and compare all available actions. That sequence separates a valid mixed best response from a randomization that is merely described by probabilities.
Check Your Understanding
For each problem, calculate expected payoffs before deciding whether a player should mix. In every payoff statement, the numbers refer to the player named in the question.
- A platform chooses Basic or Premium. Its rival chooses Open with probability 0.3 and Restricted with probability 0.7. The platform’s payoffs are Basic: 9 under Open and 3 under Restricted; Premium: 4 under Open and 7 under Restricted. Find both expected payoffs and the platform’s best response. (Answer: Basic \(=4.8\), Premium \(=6.1\); choose Premium, so mixing with positive probability on Basic is not a best response.)
- A distributor faces rival actions A, B, and C with probabilities 0.2, 0.5, and 0.3. Its payoffs are Route P: 5, 7, 1; Route Q: 6, 4, 4; Route R: 3, 6, 6, respectively. Which route is the unique best response? (Answer: P \(=4.8\), Q \(=4.4\), R \(=5.4\); Route R is the unique best response.)
- A player’s expected payoffs against the opponent’s strategy are 11 from action M, 11 from N, and 8 from O. Which actions may receive positive probability in a best response, and what must be true if the player mixes between M and N? (Answer: M and N may receive positive probability; if both are used, each yields the same maximum expected payoff of 11. O cannot receive positive probability in a best response.)
- In a two-player game, the first player mixes between East and West. Against the second player’s strategy, the first player receives 6 from either action. The second player receives 4 from East and 7 from West. Is this enough to establish that the first player’s mix is a best response and that the profile is a Nash equilibrium? (Answer: The first player is best responding among East and West, but the profile is not established as a Nash equilibrium; the second player strictly prefers West and must also be best responding.)
- A player assigns positive probability to both Safe and Bold. Against the opponent’s strategy, Safe yields 3 and Bold yields 5. What change improves expected payoff, and what does this imply about the proposed mix? (Answer: Shift probability from Safe to Bold; the proposed mix is not a best response because its support actions do not yield equal expected payoffs.)