From Outcomes to Payoffs
A strategic-form game lists each player’s available actions and assigns a payoff to each possible action profile. The payoff is not necessarily money received. It is a numerical representation of how much the player prefers the resulting outcome. Constructing a matrix therefore requires two steps: identify the outcomes associated with action profiles, then encode each player’s preferences over those outcomes.
This distinction matters because a payoff number can look more precise than the information behind it. If the only information is that one outcome is preferred to another, a ranking can represent that ordering. It does not, by itself, tell us that the preferred outcome is twice as good, or that one player’s payoff can be meaningfully compared with another’s.
For a game with several players, each action profile has one payoff for each player. The first coordinate describes the first player’s preference ordering; the second describes the second player’s ordering. Each player’s payoffs must be ranked using that player’s own preferences. A high number for one player does not automatically mean a high number for another.
Ranking Outcomes Without Inventing Cardinal Meaning
Suppose a player’s outcomes are listed in order from least preferred to most preferred. A simple encoding assigns increasing integers in that order. If two outcomes are tied, assign them the same number. For example, if \(C\) is least preferred, \(A\) and \(B\) are tied above \(C\), and \(D\) is best, the scores \(1,2,2,3\) represent the ranking. The difference between 3 and 2 is not being claimed to equal the difference between 2 and 1.
The particular numbers are usually not unique. The scores \(1,2,2,3\), \(10,40,40,100\), and \(-5,0,0,8\) all encode the same ordering. What must remain unchanged is the order and the placement of ties. If the evidence supports only a ranking, do not infer a meaningful payoff difference from the chosen score scale.
Worked Example: Encoding a Player’s Outcome Preferences
A regional supplier is choosing among four hypothetical contract outcomes. Its decision maker states that a short contract with flexible renewal is preferred to a long contract with a fixed price; those two are preferred equally to a short contract with a fixed price; and all three are preferred to a long contract with flexible renewal. Construct one ordinal payoff assignment, including the tie.
Step 1: Write the preference order. Let \(R\) denote short and flexible, \(F\) long and fixed-price, \(S\) short and fixed-price, and \(L\) long and flexible. The stated ordering is \(R \succ F \sim S \succ L\), where \(\succ\) means strictly preferred and \(\sim\) means indifferent.
Step 2: Assign increasing scores. Give the least-preferred outcome \(L\) a score of 1. Since \(S\) and \(F\) are tied and both preferred to \(L\), assign each a score of 2. Give the most-preferred outcome \(R\) a score of 3.
Step 3: Check the representation. The scores satisfy \(u(R)>u(F)=u(S)>u(L)\), exactly matching the stated preferences. A score of 3 does not imply that \(R\) is one unit better than \(F\) in an economically measurable sense; it only places \(R\) above the tied outcomes.
Method and mistake avoided. The method is to order outcomes first and assign numbers second. Giving \(F\) and \(S\) different scores would incorrectly erase the stated indifference; treating the score gap as a measured benefit would claim more than the preference information provides.
Translating Outcomes into a Payoff Matrix
If outcomes are described by monetary profits, a player who prefers more profit to less can be assigned payoffs that increase with profit. The ranks must be determined separately for each player. In particular, the same action profile can occupy different ranks for the two players, and a single action profile must not be assigned a payoff inconsistent with either player’s underlying outcome data.
Worked Example: Checking Ranks Against a Profit Table
Two hypothetical firms simultaneously choose either a discount policy \(D\) or a high-price policy \(H\). Each firm prefers higher profit to lower profit. The profit pairs, in dollars per period, are given below; the first entry is Firm 1’s profit and the second is Firm 2’s profit. Convert these outcomes into ordinal payoffs, using rank 1 for each firm’s lowest profit and rank 4 for its highest.
Step 1: Rank Firm 1’s profits. Firm 1’s four profits are \(8,12,4,6\). In increasing order they are \(4<6<8<12\), so their ranks are \(1,2,3,4\), respectively. Thus \(D,D\) receives rank 3 for Firm 1; \(D,H\) rank 4; \(H,D\) rank 1; and \(H,H\) rank 2.
Step 2: Rank Firm 2’s profits independently. Firm 2’s profits are \(6,3,12,7\). In increasing order they are \(3<6<7<12\), so their ranks are \(1,2,3,4\), respectively. Thus \(D,D\) receives rank 2 for Firm 2; \(D,H\) rank 1; \(H,D\) rank 4; and \(H,H\) rank 3.
Step 3: Put both ranks into each cell. The first payoff in each pair is Firm 1’s rank, and the second is Firm 2’s rank:
Step 4: Check the entries that are easiest to misread. At \((D,D)\), Firm 2 earns 6, which is its second-lowest profit and therefore rank 2. At \((H,H)\), Firm 2 earns 7, its third-lowest profit and therefore rank 3. The two firms’ rankings are not interchangeable.
Method and mistake avoided. Rank each column of profit information—one player at a time—then combine the ranks cell by cell. Ranking only the first firm, or sorting Firm 2’s profits in the wrong order, produces a matrix that contradicts the stated outcomes.
Why Increasing Transformations Preserve a Player’s Ordering
A transformation changes the numerical labels attached to a player’s payoffs. If the transformation is strictly increasing, it preserves every strict ordering and every tie. For deterministic outcomes, this means that whenever one action leads to a more-preferred outcome than another, the transformed numbers continue to indicate that preference. The result applies separately to each player; one player’s transformation need not be the same as another’s.
The justification is direct. If \(u(A)>u(B)\), strict increase gives \(f(u(A))>f(u(B))\). If \(u(A)=u(B)\), applying the same function to both sides gives \(f(u(A))=f(u(B))\). Conversely, if the transformed values are ordered in the opposite direction, strict increase would contradict the original ordering. Thus the transformation changes labels, not the player’s deterministic preferences.
Holding the other players’ actions fixed, the player’s preferred actions are therefore unchanged by a strictly increasing transformation of that player’s payoffs. The qualification “holding the other players’ actions fixed” matters: the comparison uses the payoff entries relevant to the player’s available choices in that situation. This statement concerns preservation of preference comparisons; it does not assert that payoff differences, total welfare, or comparisons across players are preserved.
Worked Example: Rescaling Payoffs Without Changing Individual Choices
Consider a hypothetical two-player game with actions \(A,B\) for Player 1 and \(X,Y\) for Player 2. The payoff pairs are Player 1’s payoff first and Player 2’s second. Transform Player 1’s payoff using \(u'_1=3u_1+7\), and Player 2’s using \(u'_2=2u_2-5\). Then check whether each player’s preferred action, holding the other player’s action fixed, changes.
Step 1: Transform each payoff pair. At \((A,X)\), the transformed pair is \((3(2)+7,\;2(1)-5)=(13,-3)\). At \((A,Y)\), it is \((3(5)+7,\;2(3)-5)=(22,1)\). At \((B,X)\), it is \((3(4)+7,\;2(2)-5)=(19,-1)\). At \((B,Y)\), it is \((3(1)+7,\;2(4)-5)=(10,3)\).
Step 2: Compare Player 1’s actions in each column. If Player 2 chooses \(X\), Player 1 prefers \(B\), since \(4>2\); after transformation, \(19>13\), so the preference remains \(B\). If Player 2 chooses \(Y\), Player 1 prefers \(A\), since \(5>1\); after transformation, \(22>10\), so the preference remains \(A\).
Step 3: Compare Player 2’s actions in each row. If Player 1 chooses \(A\), Player 2 prefers \(Y\), since \(3>1\); after transformation, \(1>-3\), so the preference remains \(Y\). If Player 1 chooses \(B\), Player 2 prefers \(Y\), since \(4>2\); after transformation, \(3>-1\), so the preference remains \(Y\).
Check and method. Both transformations have positive slopes, 3 and 2, so both preserve each player’s ordering. The conditional preferences match in all four comparisons. The mistake to avoid is comparing the transformed payoffs for different players as though they shared a common scale.
Strategic Comparisons Are Not Interpersonal Welfare Comparisons
Ordinal payoffs support comparisons within a player’s preferences. They do not establish that Player 1’s payoff of 4 is larger in a meaningful welfare sense than Player 2’s payoff of 3. Nor do they justify adding the two players’ payoffs and treating the sum as a measure of social welfare. Each player’s ordinal scale can be changed independently without changing that player’s ordering.
For example, a positive affine transformation can preserve every player’s preferences while changing the apparent size of payoff differences between players. An interpersonal comparison therefore requires additional assumptions: the analyst must specify a common, comparable scale and justify how differences in one person’s measured utility relate to differences in another’s. Those assumptions are not supplied by an ordinary payoff matrix.
There is also an important limit to the transformation result. For deterministic outcomes, any strictly increasing transformation preserves the ordering. If payoffs are used to evaluate lotteries by expected utility, however, an arbitrary nonlinear increasing transformation can change the ranking of lotteries. Positive affine transformations preserve expected-utility comparisons because, for \(a>0\),
Multiplication by a positive number and addition of a constant preserve the ordering of expected values. A general nonlinear transformation does not distribute through expectation in this way. Thus, when a model includes random outcomes or mixed choices evaluated by expected payoff, positive affine transformations—not every strictly increasing transformation—preserve the expected-utility ordering.
Worked Example: Why Independent Payoff Scales Do Not Measure Social Welfare
Two hypothetical outcomes have payoff pairs \((2,1)\) and \((1,4)\). The first coordinate represents Player 1’s ordinal payoff and the second represents Player 2’s. A reviewer proposes ranking the outcomes by the sum of the two payoffs. Show how that ranking can reverse after valid positive rescalings of the individual players’ payoffs.
Step 1: Add the original numbers. The first outcome has sum \(2+1=3\). The second has sum \(1+4=5\). On this scale, the second outcome has the larger sum.
Step 2: Rescale each player separately. Use \(u'_1=10u_1\) for Player 1 and \(u'_2=0.1u_2\) for Player 2. Both transformations have positive slopes, so neither changes its player’s ordinal ranking.
Step 3: Recalculate the sums. The first outcome becomes \((20,0.1)\), with sum \(20+0.1=20.1\). The second becomes \((10,0.4)\), with sum \(10+0.4=10.4\). The first outcome now has the larger sum.
Conclusion and check. The individual preference orderings were preserved, but the ranking by summed payoffs reversed. Therefore the original sum did not follow from ordinal preferences alone. To interpret a total as a social-welfare measure, one must add assumptions that make the players’ payoff scales interpersonally comparable.
Common Mistakes to Avoid
- Using inconsistent ranks. Each matrix entry must agree with the underlying outcome and that particular player’s preference ordering.
- Reading rank gaps as measured benefits. Ordinal scores show which outcome is preferred, not how much more it is preferred.
- Applying the same ranking to every player. Players can order the same outcomes differently; rank each player’s payoffs separately.
- Confusing individual choice preservation with social-welfare preservation. Positive rescaling can preserve each player’s preferences while reversing comparisons based on adding players’ payoffs.
- Using a nonlinear transformation for lottery comparisons without checking the model. Strictly increasing transformations preserve deterministic rankings, but expected-utility comparisons over lotteries are guaranteed to be preserved by positive affine transformations.
Check Your Understanding
For each problem, distinguish the information represented by the payoff numbers from conclusions that require additional assumptions.
- A decision maker ranks four outcomes \(M \succ N \sim P \succ Q\). Assign integer ordinal payoffs starting at 1 for the least-preferred outcome, using one score for each distinct preference level. (Answer: \(u(Q)=1,\;u(N)=u(P)=2,\;u(M)=3\).)
- A player’s payoffs for outcomes \(R,S,T\) are \(2,5,8\), respectively. Apply \(f(u)=4u-6\), list the transformed payoffs, and state whether the ordering changes. (Answer: \(R=2,\;S=14,\;T=26\); the ordering remains \(T\succ S\succ R\).)
- In a two-action choice, a player receives payoff 7 from action \(C\) and 3 from action \(D\), with the other player’s action held fixed. Transform payoffs using \(u'=0.5u+9\). Which action remains preferred, and what are the transformed values? (Answer: \(C\) remains preferred; the values are \(12.5\) for \(C\) and \(10.5\) for \(D\).)
- Two outcomes have payoff pairs \((3,2)\) and \((2,6)\). Compare their original payoff sums, then transform the first player’s payoffs by \(u'_1=5u_1\) and the second player’s by \(u'_2=0.2u_2\). What are the new sums, and what does the reversal show? (Answer: Original sums are 5 and 8; transformed sums are 15.4 and 11.2, so the ranking reverses and the ordinal numbers alone do not determine interpersonal welfare.)
- A lottery gives utility 0 or 10 with equal probability; another outcome gives utility 4 for certain. Compare expected utilities before and after applying the strictly increasing transformation \(f(u)=\sqrt{u}\). (Answer: Originally the lottery has expected utility 5 versus 4 for the certain outcome, so the lottery ranks higher; after transformation the lottery has expected utility \((0+\sqrt{10})/2\approx1.5811\) versus \(2\) for the certain outcome, so the certain outcome ranks higher. A nonlinear increasing transformation need not preserve lottery rankings.)