Identify who chooses, what they know, and how joint choices produce outcomes.
By the end: build a small game without confusing material outcomes, preferences, and predictions.
This course assumes arithmetic and percentages, not economics, calculus, or a previous course. It introduces probability and utility where needed. All payoffs, types, bids, and rules are constructed examples, not estimates of actual preferences or advice for a real negotiation.
Choices depend on other choices
Two neighbors share a pump. Each can maintain it or skip maintenance. A decision is strategic when its consequences depend on another decision-maker's choice. Contrast this with choosing an umbrella given an exogenous rain forecast. Rain does not respond to your incentives; the neighbor might.
A game specifies players, available actions, timing, information, and payoffs. Here both neighbors choose without observing the other's current action. Simultaneous means this information condition, not necessarily that their hands move at the same instant. The four cells below list every joint action.
Specify a choice before predicting it
A player is someone whose choice the model represents. The pump is not a third player unless it makes a modeled choice. Weather can affect the pump without being a player. An action is an available choice, such as Maintain. An outcome is what follows from all the choices, such as a working pump plus somebody's maintenance effort. Keeping these objects separate prevents a common mistake. A neighbor can choose effort, but cannot directly choose the joint outcome 'both maintain.' That outcome also requires the other neighbor's action.
A strategy is a complete rule for choosing wherever the player may have to act. In this one-decision example, each strategy is simply one action. A strategy profile lists one strategy for each player. Maintain, Skip is one profile, with Row's choice first. Four profiles result because either of Row's two actions can accompany either of Column's two actions. Listing all four is a completeness check. If you cannot explain an outcome for one combination, the proposed model leaves part of the interaction unspecified rather than merely difficult to predict.
Two independent action lists produce four profiles
Row / Column
Maintain
Skip
Maintain
Row Maintain; Column Maintain
Row Maintain; Column Skip
Skip
Row Skip; Column Maintain
Row Skip; Column Skip
Rows change Row's choice; columns change Column's choice. Each cell specifies one complete profile, not one person's action.
Suppose Row submits a maintenance decision on Monday and Column submits on Tuesday. If Column cannot inspect Monday's submission, neither can condition their decision on the other's actual choice. The matrix still represents the information available for choosing. If Column sees Row's submission first, Column can instead use a response rule such as maintain after maintenance and skip after skipping. That is a different information structure, even if the final physical pump outcomes are unchanged. The later tree lessons will represent those conditional opportunities explicitly rather than adding them silently to this matrix.
For a separate paper construction, imagine each neighbor starts with one utility unit of baseline benefit. Any maintenance raises each neighbor's benefit to five, but costs its provider two utility units. Two maintainers therefore each receive five minus two, giving three. A lone maintainer gets three while the skipper gets five. Two skippers each retain one. This plausible story does not reproduce the explorer, whose lone maintainer receives zero. A story about a shared pump cannot determine the numbers without further assumptions about how effort and failures affect each person.
To reproduce the explorer with a paper accounting rule, instead assign gross benefits five for two maintainers, two to a lone maintainer, five to the neighbor relying on that maintainer, and one for mutual skipping. Subtract a cost of two from each maintainer only. The four resulting pairs are three and three, zero and five, five and zero, and one and one. This accounting is stipulated utility arithmetic, not a measurement of water output. Its asymmetric gross benefits could represent reliability or access differences, but that interpretation would need evidence in an actual pump study.
Subtract each maintainer's cost from that person's gross utility
Profile
Row gross − cost
Column gross − cost
Net pair
Maintain, Maintain
5 − 2 = 3
5 − 2 = 3
3, 3
Maintain, Skip
2 − 2 = 0
5 − 0 = 5
0, 5
Skip, Maintain
5 − 0 = 5
2 − 2 = 0
5, 0
Skip, Skip
1 − 0 = 1
1 − 0 = 1
1, 1
Paper accounting reproduces the preset but does not measure pump output. Only maintainers incur the stipulated two-unit cost.
Use the distinction to inspect an unfamiliar claim. First ask what each person can choose, then what combinations are possible, then how each combination affects each person's preferences. Do not infer a missing action from an attractive outcome. A maintenance rota, inspection service, or option to leave could be relevant, but none is an action in this two-choice explorer. Adding one would require a larger model. The matrix deliberately holds those possibilities outside its scope so that comparisons among the stated choices remain precise and can be checked one coordinate at a time.
Read a payoff pair
The first number belongs to the row player and the second to the column player. In this constructed example, mutual maintenance gives each 3 utility units. Skipping while the other maintains gives the skipper 5 and the maintainer 0. Mutual skipping gives each 1. Larger numbers mean more preferred outcomes for that player.
A payoff is a representation of preferences over the full consequences, not automatically money. It can include effort, safety, fairness, or concern for the other person. Do not compare two people's utility levels as if they shared a physical unit. Adding their payoffs requires an extra social objective and a justified common scale.
Translate consequences into a preference ranking
Utility is a numerical representation of a person's preferences. In a comparison of certain outcomes, only that person's ordering matters. Row prefers five to three, three to one, and one to zero in the maintenance preset. The numbers do not establish how much water Row receives or how strongly Column feels about anything. A utility of zero is a chosen reference point, not proof of no benefit or no suffering. Negative numbers can represent less preferred outcomes without implying debt. Each player's scale needs its own interpretation before using the entries as evidence about a real decision.
Consider a paper variant in which Row cares about Column's outcome as well as Row's own baseline utility. Stipulate that Row's revised utility equals Row's original entry plus half of Column's original entry. Mutual maintenance then gives Row three plus half of three, or four and a half. Skipping against maintenance gives five plus half of zero, or five. Maintaining against skipping gives zero plus half of five, or two and a half. Mutual skipping gives one plus half of one, or one and a half. These are revised preferences, not extra water.
Concern for Column changes one of Row's action comparisons
Column action
Row Maintain
Row Skip
Row prefers
Maintain
3 + 0.5 × 3 = 4.5
5 + 0.5 × 0 = 5
Skip
Skip
0 + 0.5 × 5 = 2.5
1 + 0.5 × 1 = 1.5
Maintain
This paper preference rule adds half of Column's baseline utility. It changes Row's ranking, unlike a scale relabeling.
This preference change has different effects against different actions. Against a maintaining Column, Row still prefers skipping, since five exceeds four and a half. Against a skipping Column, Row now prefers maintaining, since two and a half exceeds one and a half. Concern for another person need not turn every comparison toward the same action. Nor does observing maintenance identify concern as its cause. A penalty, mistaken belief, or different effort cost could generate that choice too. The paper example isolates one explanation by stating the preference rule before calculating its consequences.
Now distinguish a preference change from a harmless relabeling. In a second paper comparison, multiply every original Row payoff by ten and add seven. Row's four entries become thirty-seven, seven, fifty-seven, and seventeen in matrix order. Every comparison between Row's certain outcomes stays the same. Nothing in Column's scale needs to change. It would be wrong to conclude that Row suddenly matters ten times as much to society. The numbers changed because we chose a different representation, while Row's ranking and the available physical outcomes remained fixed throughout this comparison.
Adding the two players' entries can therefore give a misleading social ranking. Originally, mutual maintenance has a sum of six and Row skipping against maintenance has a sum of five. After only Row's relabeling, those sums become forty and fifty-seven. The apparent ranking by totals reverses even though neither person's ordering changed. This does not make every social comparison impossible. Both people prefer three and three to one and one. That comparison uses each person's own preference direction. Choosing weights for a combined welfare measure would require a separate justification beyond the game matrix.
Relabeling Row's utility reverses a ranking by sums
Profile
Original pair; sum
Relabeled pair; sum
Maintain, Maintain
3, 3; total 6
37, 3; total 40
Skip, Maintain
5, 0; total 5
57, 0; total 57
Maintain, Skip
0, 5; total 5
7, 5; total 12
Skip, Skip
1, 1; total 2
17, 1; total 18
Row's relabeling preserves each individual ranking. The reversed totals show why an unjustified interpersonal sum is not invariant.
The later expected-utility comparisons will need stronger information about utility differences because they average possible utilities. For now, keep the task narrow. The explorer supplies stipulated numbers and asks what follows from them. It does not estimate preferences from your selections or convert material costs into utility. If someone reports cash amounts instead, ask whether larger own cash always represents the person's full preference. Fairness, effort, risk, and consequences for another person may matter. A model can include those concerns, but must state how they affect the comparisons rather than presume a cash-only objective.
Compare incentives and outcomes
Inspect one action change at a time
Shared maintenance. Each cell lists Row, then Column utility.
Row / Column
Maintain
Skip
Maintain
Row: 3Column: 3
Row: 0Column: 5 ✓
Skip
Row: 5 ✓Column: 0
Row: 1 ✓Column: 1 ✓Both best responses
✓ means that player cannot improve by switching alone while the other action stays fixed. Ties receive marks too. A border identifies the selected cell.
Selected payoffs: 3, 3. Nash equilibrium: No. Pareto efficient among pure cells: Yes.
Unilateral comparison
Stay
Switch alone
Row; Column holds Maintain
3
5
Column; Row holds Maintain
3
5
Player
Strictly dominant action
Weakly dominant action
Row
Skip
Skip
Column
Skip
Skip
Under our definition, strictly dominant actions also satisfy weak dominance.
A game preset replaces the payoffs and resets both actions to the first listed choice. Action buttons change only that player’s choice. Utilities are stipulated. Pareto efficiency below compares only the four pure cells, not lotteries or outside options.
Joint action
Row utility
Column utility
Maintain, maintain
3
3
Skip, maintain
5
0
Read each player's entry separately; the larger entry is not a social verdict.
Move one action while holding the other fixed
Start the explorer with Shared maintenance and select Row Maintain, Column Maintain. The selected pair is three and three. Keep Column at Maintain and press Row Skip. The new pair is five and zero. Row's utility rose by two while Column's fell by three. This is a unilateral change, meaning one player changed their own action while the other player's action stayed fixed. The result is not a prediction that Column will accept the change, forgive it, or respond later. Those possible responses are not additional choices inside this single comparison.
Next hold Row at Skip and press Column Skip. The pair becomes one and one. Relative to the previous cell, Column gains one and Row loses four. If you now press Row Maintain, you reach zero and five. These moves trace three edges of the four-cell matrix. A diagonal change would require changing both players' actions, so it cannot answer what one player can accomplish alone. The diagram shows this geometry as a map of feasible action changes. Its distances are schematic, not sizes of utility gains or probabilities of behavior.
An action change moves along an edge, not a diagonal
These maintenance moves hold the other action fixed. Arrow labels give the changing player's gain; positions are schematic.
Return to mutual skipping and compare it with mutual maintenance. Both would gain two utility units from the joint change. Yet if Column stays at Skip, Row's attempt to maintain alone lowers Row's utility from one to zero. This is why a desirable shared destination does not automatically provide an individual route to it. The claim 'Row can improve the outcome' needs a specified meaning. Row can change Row's action, propose a joint change, or perhaps buy an enforceable arrangement in another model. These are different opportunities with different incentive calculations.
The check marks next to utilities identify best responses. A best response gives the player the highest utility available against the opponent's fixed action. With Column maintaining, Row's five receives a mark rather than Row's three. With Column skipping, Row's one receives a mark rather than Row's zero. Column's checks compare Column's entries across a row, not down a column. The current selection border and the checks answer different questions. The border says which profile you selected. The checks say which individual choices resist improvement under the fixed opposing action.
The status also reports Nash equilibrium and Pareto efficiency. Here Nash means that neither player can improve by changing alone. Mutual Skip passes because either lone maintainer would get zero instead of one. Pareto efficiency asks whether another allowed joint outcome improves somebody without hurting anybody. Mutual Skip fails because mutual Maintain gives both three instead of one. The explorer compares only its four pure cells, where pure means an action chosen with certainty. These labels can disagree without any arithmetic error because one checks individual opportunities and the other checks joint alternatives.
Mutual skipping passes the unilateral test but fails the joint test
Change from Skip, Skip
New pair
Row change
Column change
Row alone maintains
0, 5
−1
+4
Column alone maintains
5, 0
+4
−1
Both maintain
3, 3
+2
+2
The same starting pair permits no individual improvement, yet another pure cell improves both players. These tests differ.
Choose Joint project next and observe that changing the game resets both selections to the first listed actions, Join and Join. Select Row Safe while leaving Column at Join. Row now gets four rather than six, so this move loses two. Keep Row Safe and change Column to Safe. Column gets four rather than zero, a gain of four. Returning Row to Join would give Row zero rather than four. The controls change selections, not payoff entries. Comparing this sequence with maintenance shows that the same four-cell format can express very different dependencies between choices.
State what the simplification omits
Changing preferences can change the game even when the material pump outcomes stay fixed. If maintenance also earns a valued reputation or avoids a penalty, that consequence belongs in the payoff. If a neighbor can move away, an omitted action may invalidate the proposed solution.
A matrix is a model specification. It does not establish that the neighbors know these numbers, optimize them, or share correct beliefs. Begin a real analysis by asking which consequences, available choices, and information would justify the entries. Keep a prediction separate from the diagram that represents the assumptions.
Audit the assumptions that support a matrix
A payoff table makes conditional statements. If these actions occur under these preferences, this pair follows. To turn that into a forecast, an analyst must add reasoning about choices and evidence about the setting. Knowing your own preferences does not establish that you know the neighbor's. Even knowing both rankings does not prove that either person understands the available actions. Separate a mathematical implication of the stated model from an empirical claim about a person. A selected cell in the explorer provides the former kind of illustration, not new observations of anyone's maintenance behavior.
Check the action set before refining the utilities. In a separate paper extension, suppose either neighbor can hire an outside service that guarantees their own utility of two regardless of the other's choice. This third action changes the comparison against Skip. Maintaining gives zero, skipping gives one, and hiring gives two. The original conclusion that Skip is the best available response no longer holds in that situation. The original arithmetic was correct for its two actions. The application failed because a feasible alternative was missing. This extension is not an editable option in the explorer.
An omitted action can beat both displayed responses
Row action against Column Skip
Original choices
With hiring available
Maintain
0
0
Skip
1, best available
1
Hire
Not available
2, best available
Paper extension only. Hiring guarantees utility two for the hiring player; the explorer has no such third action.
Check consequences for people outside the modeled pair as well. A service might charge somebody else, draw water away from another household, or require unpaid work. The pair of utilities does not record those effects unless the model includes them in the neighbors' preferences, and even then it does not directly represent the outsiders' own preferences. A favorable comparison for both neighbors is therefore a claim about these two modeled players. Expanding a policy judgment to everyone affected needs additional information about whose outcomes change and how those outcomes should be assessed.
Uncertain numbers can sometimes support a firm conclusion without a precise estimate. In a paper sensitivity check, suppose Row's utility from maintaining against Maintain lies between two and four, while skipping against Maintain stays five. Skipping remains better throughout that range because even four is below five. Against Skip, let maintaining lie between zero and two while skipping stays one. Now the ranking is uncertain. At zero maintenance loses, at one it ties, and at two it wins. Report which conclusion survives the allowed uncertainty instead of treating every entry as equally secure.
Changing the information can matter even when no utility entry changes. If a trustworthy inspection reveals the neighbor's action before your choice, you can select separately for each observed action. If the inspection is unreliable, its accuracy becomes part of a new decision problem. Neither case follows merely from writing the same four payoff pairs. The present matrix has no inspection control or observation history. Later lessons separate hidden information from observed moves. Until then, distinguish uncertainty about a payoff entry from uncertainty about which opponent action will accompany your choice.
Some rankings survive an interval of uncertain payoffs
Paper sensitivity check. The first interval never reaches Skip's payoff; the second crosses it and leaves the ranking unresolved.
A useful model report states what would falsify its application. If maintenance requires joint physical access, the lone-maintainer cells may describe impossible actions. If an enforceable obligation already exists, the skipping utilities may omit a penalty. If the neighbors value maintaining a norm, the preference ranking may differ from the stipulated one. These are specific reasons to revise a game, not reasons to discard the method. Once the actions, information, and preferences are defensible, the same discipline applies to a new setting. Enumerate combinations first, then test what one player can actually change.
Exercises
Read the other person's payoff
Row maintains and Column skips. What does each receive? Can Row make the outcome 3,3 by changing only Row's action?
Hint
Find the current cell, then change its row while retaining the column.
Worked answer
Shared maintenance. Each cell lists Row, then Column utility.
Row / Column
Maintain
Skip
Maintain
Row: 3Column: 3
Row: 0Column: 5 ✓
Skip
Row: 5 ✓Column: 0
Row: 1 ✓Column: 1 ✓Both best responses
✓ means that player cannot improve by switching alone while the other action stays fixed. Ties receive marks too. A border identifies the selected cell.
Row choice
Column held at Skip
Payoffs
Maintain
Skip
0,5
Skip
Skip
1,1
A unilateral move changes one coordinate only.
The current payoff is 0 for Row and 5 for Column. Row alone can switch to Skip, reaching 1,1. Reaching 3,3 also requires Column to switch to Maintain. A joint improvement is not automatically available as a unilateral choice.
Money is not the whole preference
A person prefers receiving 6 while a partner receives 6 over receiving 8 while the partner receives 0. Does this violate rationality in the game-theory sense?
Hint
Ask what outcomes the person ranks, not whether they maximize their own cash.
Worked answer
Outcome
Own money
Partner money
Stated ranking
Shared
6
6
Preferred
Unequal
8
0
Less preferred
Material payoffs and utility rankings are different data.
No. A consistent preference can include the partner's outcome. A model that equates utility with own money would misrepresent this person. You need their ranking or a defensible utility specification; do not infer irrationality from disagreement with an assumed cash-only objective.
Sources and further reading
Selected reference passages support the definitions and algorithms below. The games, explanations, and implementations are original. Terminology and assumptions are stated within each lesson; source review is not an independent audit of this course. Optional links are not runtime dependencies.
Selected transcript passages on players, strategies, preferences, and how alternative payoffs change the game. Examples here are original, not observations of learner behavior.