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---

### Executive summary
- Two bank resolution policies compared via simulations using an Abelian sand pile (chip-firing) metaphor:
  - laissez faire: supervisors do not intervene; shareholders absorb failing banks.
  - managed (supervisors’ directed mergers): supervisors direct the strongest institutions (“white knights”) to absorb failing banks via P&A-style operations (each takeover equals ¼ of the failing bank’s balance sheet and reduces the white knight’s capital by 1 unit).
- Main high-level findings:
  - Crises (welfare-loss events) occur less frequently under the managed strategy but are, on average, more costly per event.
  - Over a medium/long term simulation horizon, total social welfare loss is typically lower in the managed case than in laissez faire.
  - The optimal network structure (minimizing average welfare loss per failing institution or frequency of failing events) depends on the chosen resolution policy and supervisors’ objectives.

### Model, assumptions, and scenarios
- Network representation:
  - Directed graph of banks as nodes; links represent shareholder (ownership) relations.
  - Number of nodes in primary simulations: 15.
  - Directed graph built from an asymmetric square matrix (15 x 15) with zeros on the diagonal.
  - Simulations also analyze all possible networks of size 4 (2^12 = 4096 possibilities).
- Capital, thresholds, and shocks:
  - Initial random capital endowment per bank: min of 8 and max of 12 capital units.
  - Prudential minimum capital required: 8 units.
  - Failure threshold: banks fail when accumulated losses exceed 4 (i.e., go below the 8 minimum).
  - Each unexpected NPL shock reduces a bank’s capital by 1 unit; each NPL equals one unit.
- Shareholder loss allocation rules:
  - When a controlled bank fails, each shareholder is allocated ¼ of the failing bank balance sheet and must absorb 1 unit of capital loss.
  - If there are more than 4 shareholders, supervisors randomly select which shareholders absorb losses.
  - When shareholders are less than 4 (or insufficient capital), the unabsorbed portion is liquidated generating a social welfare loss (a “crisis”).
- Two primary simulation scenarios:
  - Laissez faire: shareholders absorb failing banks as above; no supervisor-directed reallocation beyond the shareholder mechanism.
  - Managed: supervisors direct losses to the strongest nodes (those with highest remaining capital) to act as white knights; each P&A reduces a white knight’s capital by 1 unit and white knights initially must have capital at or above 9 to be eligible.
- Additional scenario:
  - Cynical supervisors: supervisors manage failures by directing losses to strong banks but their objective in network optimization is to maximize time between crises (minimize frequency), not total welfare loss.

### Simulation design and parameters
- Typical simulation horizons and repetitions:
  - Primary runs: 300 iterations (interpreted as “days”).
  - For 4-node exhaustive analysis: each network exposed to the same set of 100 shocks.
  - Genetic algorithm (GA) optimization on 15-node networks: population size 26 individuals per generation, 30 generations for the main GA runs (26 x 30 = 780 networks evaluated per experiment).
  - Extended robustness runs: three sets of 1000 simulations for the three bank resolution policies; each of the 1000 simulations runs over 10 generations.
- Network feature measures used in analysis:
  - Size: total number of links/relationships across institutions.
  - Average clustering coefficient: tendency of nodes to cluster (probability banks have common shareholders).
  - Average minimum path: average number of steps to link any two nodes (measure of interconnectedness / “completeness”).

### Key simulation outcomes and quantitative results
- Managed vs. laissez faire (representative observations):
  - Frequency: crises occur less frequently under managed allocation.
  - Severity: average welfare loss per crisis is larger under managed allocation because failures of white knights trigger broader domino effects.
  - Total welfare loss over 300 “days”:
    - Managed total loss typically lower than laissez faire total loss (Figure 6 shows “tot Loss man” lower than laissez faire over full period).
- 4-node exhaustive experiments:
  - Laissez-faire champion: one configuration with average loss slightly over 3.5 units per crisis; crisis occurs on average every 2.37 “days”; total loss over 100 days = 147.68.
  - Managed champions: 64 configurations with minimum average loss of 7.1 units per crisis; crisis occurs on average every 4.84 “days”; total loss over 100 days = 146.69.
  - Conclusion: managed regime yields more networks that achieve the minimum total loss, increasing probability of achieving a best outcome.
- Extended GA and large-sample experiments (1000 simulations each):
  - Total loss distribution (Figure 11):
    - Managed case concentrated: 947 cases out of 1000 concentrated at outcome 2046 (indicating GA found similar optimal configurations repeatedly).
    - Laissez faire average total loss ≈ 2055.5.
    - Cynical supervisors average total loss ≈ 2063 (more volatile; best-case occasionally comparable to managed, worst-case often worse than laissez faire).
- Effect of excluding symmetric links (no direct cross-holdings):
  - Managed case: total losses remain lower than laissez faire; minimum Total Loss for managed = 137 and for laissez faire = 141.
  - Removing symmetric links tends to:
    - Increase frequency of crises under laissez faire (shorter intervals).
    - Decrease frequency of crises under managed rule, especially for cynical supervisors (longer intervals).
    - Not materially change total welfare loss in many cases (total capital in system unchanged), but changes crisis frequency and average crisis severity.
- Strongly connected vs not-strongly connected networks (Table 1):
  - Laissez Faire:
    - not-connected Ave SIZE 62.74 vs connected 68.12
    - Ave CLUSTER 0.30 vs 0.30
    - Ave TOT LOSS 2055.25 vs 2055.34
  - Managed:
    - not-connected Ave SIZE 64.54 vs connected 73.86
    - Ave CLUSTER 0.31 vs 0.31
    - Ave TOT LOSS 2046.00 vs 2046.02
  - Cynical Supervisors:
    - not-connected Ave SIZE 60.22 vs connected 61.54
    - Ave CLUSTER 0.29 vs 0.29
    - Ave TOT LOSS 2063.89 vs 2064.47
  - Conclusion: presence of isolated subnetworks alters size and clustering but does not substantially change average total welfare loss across regimes.

### Network optimization and structural insights
- Rugged performance landscape:
  - For 4-node networks, the landscape of average welfare loss is highly non-linear and rugged; small changes in links can produce large shifts in performance.
  - For larger networks (15 nodes), GA used to search for better-performing topologies; 780 examined networks is a small subset of all 2^210 possible structures.
- Which structural features matter (pooled regressions and experiments):
  - No single universal architecture minimizes losses across all regimes.
  - Laissez faire: larger number of shareholding links (network size) helps reduce average loss under laissez faire.
  - Managed regime (minimizing total welfare losses) benefits from networks that are:
    - Larger in total number of shareholding relationships,
    - More closely connected (lower average minimum path),
    - More clustered (higher clustering coefficient).
  - Cynical supervisors (minimizing time between crises): networks that minimize frequency of crises often coincide with those that minimize total welfare losses across regimes, but cynical supervisors’ optimization yields more volatile outcomes and frequently worse total loss.
- Symmetric links:
  - Disallowing direct reciprocal cross-holdings (A→B and B→A) preserves many results and can be beneficial under active supervisory strategies by increasing time between crises without automatically increasing total welfare loss.

### Policy implications and recommendations
- Managed (supervisor-directed merger/P&A) strategies:
  - Tend to reduce total social welfare loss over time despite increasing the average cost per crisis event; simulations support the view that directing mergers is, on net, welfare-improving relative to laissez faire in the modeled environment.
  - May create larger systemic institutions whose failure produces larger domino effects; supervisors should account for systemic buildup when designing resolution frameworks.
- Regulatory treatment of cross-holdings:
  - Prohibiting or netting reciprocal cross-holdings (symmetric links) is generally beneficial under an active supervisory regime: it increases the period between crises and helps supervisors identify stronger white knights more reliably.
- Supervisory objective design:
  - If supervisors optimize for minimizing frequency of crises (politically motivated or “cynical” objective), the resulting networks may produce longer time between crises but often higher total welfare losses and more volatile outcomes; supervisors’ objective functions matter for desirable network architecture.
- Monitoring and macroprudential design:
  - Regulators should consider how resolution practices interact with network topology; possible regulatory actions (e.g., restrictions on cross-holdings, disclosure of ownership networks, capital requirements calibrated to network position) could influence systemic resilience.
- Suggested extensions (simulation/modeling):
  - Use weighted directed graphs to capture magnitude of ownership/control relations.
  - Disallow triangular cross-holdings (A→B, B→C, C→A) to more strictly prevent complex cross-ownership structures.
  - Experiment with higher capital thresholds than the maintained 8 to evaluate nonlinear effects on frequency and total welfare loss.
  - Explore recapitalization and entry dynamics (model currently assumes no new capital creation).

### Annex Tables — OLS summaries (selected coefficients and diagnostics)
- Annex 1 — Laissez Faire, Managed, and Cynical supervisors cases: OLS results for Average loss on Network size, Average cluster coefficient, and Average Shortest Path
  - Managed case (Dependent Variable: LOSS_MAN; Method: Least Squares)
    - Sample: 1 1000; Included observations: 1000
    - Estimated equation: LOSS_MAN = C(1) + C(2)*SIZE_MAN + C(3)*CLUST_MAN + C(4)*PATH_MAN
    - Coefficients:
      - C(1) -3.562400; Std. Error 11.28470; t-Statistic -0.315684; Prob. 0.7523
      - C(2) 0.025426; Std. Error 0.039429; t-Statistic 0.644853; Prob. 0.5192
      - C(3) 1.705608; Std. Error 3.944882; t-Statistic 0.432360; Prob. 0.6656
      - C(4) 18.26081; Std. Error 5.315314; t-Statistic 3.435510; Prob. 0.0006
    - Goodness of fit and diagnostics:
      - R-squared 0.067813; Adjusted R-squared 0.065005
      - Mean dependent var 27.52884; S.D. dependent var 2.165686
      - S.E. of regression 2.094114; Sum squared resid 4367.770
      - Akaike info criterion 4.320130; Schwarz criterion 4.339761; Hannan-Quinn criter. 4.327591
      - Log likelihood -2156.065
      - F-statistic 24.15164; Prob(F-statistic) 0.000000
      - Durbin-Watson stat 1.976272
  - Laissez Faire case (Dependent Variable: LOSS_LF; Method: Least Squares)
    - Date: 08/28/24; Sample: 1 1000; Included observations: 1000
    - Estimated equation: LOSS_LF = C(1) + C(2)*SIZE_LF + C(3)*CLUST_LF + C(4)*PATH_LF
    - Coefficients:
      - C(1) 23.6977; Std. Error 2.625495; t-Statistic 9.025997; Prob. 0.0000
      - C(2) -0.022957; Std. Error 0.008981; t-Statistic -2.556009; Prob. 0.0107
      - C(3) -2.472345; Std. Error 0.824850; t-Statistic -2.997329; Prob. 0.0028
      - C(4) -5.659769; Std. Error 1.229812; t-Statistic -4.602140; Prob. 0.0000
    - Goodness of fit and diagnostics:
      - R-squared 0.035325; Adjusted R-squared 0.032419
      - Mean dependent var 11.49479; S.D. dependent var 0.439353
      - S.E. of regression 0.432173; Sum squared resid 186.0260
      - Akaike info criterion 1.164008; Schwarz criterion 1.183639; Hannan-Quinn criter. 1.171469
      - Log likelihood -578.0041
      - F-statistic 12.15726; Prob(F-statistic) 0.000000
      - Durbin-Watson stat 2.095158
  - Cynical supervisors case (Dependent Variable: LOSS_MH; Method: Least Squares)
    - Sample: 1 1000; Included observations: 1000
    - Estimated equation: LOSS_MH = C(1) + C(2)*SIZE_MH + C(3)*CLUST_MH + C(4)*PATH_MH
    - Coefficients:
      - C(1) 30.15527; Std. Error 3.972491; t-Statistic 7.591022; Prob. 0.0000
      - C(2) -0.004064; Std. Error 0.014649; t-Statistic -0.277464; Prob. 0.7815
      - C(3) 1.114338; Std. Error 1.444334; t-Statistic 0.771524; Prob. 0.4406
      - C(4) 2.239361; Std. Error 1.807025; t-Statistic 1.239253; Prob. 0.2155
    - Goodness of fit and diagnostics:
      - R-squared 0.015869; Adjusted R-squared 0.012904
      - Mean dependent var 33.72440; S.D. dependent var 0.856391
      - S.E. of regression 0.850848; Sum squared resid 721.0458
      - Akaike info criterion 2.518824; Schwarz criterion 2.538455; Hannan-Quinn criter. 2.526286
      - Log likelihood -1255.412
      - F-statistic 5.353304; Prob(F-statistic) 0.001167
      - Durbin-Watson stat 2.077381

- Annex 2 — OLS on combined sample: Average Loss on network size, clustering coefficient, and average minimum path
  - Combined Sample, OLS: Average Loss on network size (Dependent Variable: AVELOSS)
    - Date: 10/12/24; Sample: 1 3000; Included observations: 3000
    - Estimated equation: AVELOSS = C(4) + C(1)*D1*SIZE + C(2)*D2*SIZE + C(3)*D3*SIZE
    - Coefficients:
      - C(4) 27.64821; Std. Error 0.401081; t-Statistic 68.93423; Prob. 0.0000
      - C(1) -0.152971; Std. Error 0.003826; t-Statistic -39.98229; Prob. 0.0000
      - C(2) -0.001455; Std. Error 0.003814; t-Statistic -0.381618; Prob. 0.7028
      - C(3) 0.059399; Std. Error 0.003954; t-Statistic 15.02062; Prob. 0.0000
    - Goodness of fit and diagnostics:
      - R-squared 0.973651; Adjusted R-squared 0.973625
      - Mean dependent var 24.24934; S.D. dependent var 9.467683
      - S.E. of regression 1.537591; Sum squared resid 7083.105
      - Akaike info criterion 3.699644; Schwarz criterion 3.707652; Hannan-Quinn criter. 3.702524
      - Log likelihood -5545.466
      - F-statistic 36903.21; Prob(F-statistic) 0.000000
      - Durbin-Watson stat 1.893216
  - Combined Sample, OLS: Average Loss on Clustering coefficient (Dependent Variable: AVELOSS)
    - Date: 10/12/24; Sample: 1 3000; Included observations: 3000
    - Estimated equation: AVELOSS = C + C(1)*D1*CLUST + C(2)*D2*CLUST + C(3)*D3*CLUST
    - Coefficients:
      - C 27.05958; Std. Error 0.366463; t-Statistic 73.83979; Prob. 0.0000
      - C(1) -30.96315; Std. Error 0.735540; t-Statistic -42.09579; Prob. 0.0000
      - C(2) 0.864547; Std. Error 0.732593; t-Statistic 1.180120; Prob. 0.2380
      - C(3) 13.54831; Std. Error 0.752456; t-Statistic 18.00545; Prob. 0.0000
    - Goodness of fit and diagnostics:
      - R-squared 0.972738; Adjusted R-squared 0.972711
      - Mean dependent var 24.24934; S.D. dependent var 9.467683
      - S.E. of regression 1.564001; Sum squared resid 7328.515
      - Akaike info criterion 3.733704; Schwarz criterion 3.741713; Hannan-Quinn criter. 3.736585
      - Log likelihood -5596.557
      - F-statistic 35634.00; Prob(F-statistic) 0.000000
      - Durbin-Watson stat 1.884734
  - Combined Sample, OLS: Average Loss on Average Minimum Path (Dependent Variable: AVELOSS)
    - Date: 10/12/24; Sample: 1 3000; Included observations: 3000
    - Estimated equation: AVELOSS = C + C(1)*D1*PATH + C(2)*D2*PATH + C(3)*D3*PATH
    - Coefficients:
      - C 18.10670; Std. Error 0.824059; t-Statistic 21.97257; Prob. 0.0000
      - C(1) -4.371290; Std. Error 0.545530; t-Statistic -8.012918; Prob. 0.0000
      - C(2) 6.249760; Std. Error 0.546427; t-Statistic 11.43750; Prob. 0.0000
      - C(3) 10.16341; Std. Error 0.536881; t-Statistic 18.93047; Prob. 0.0000
    - Goodness of fit and diagnostics:
      - R-squared 0.979270; Adjusted R-squared 0.979249
      - Mean dependent var 24.24934; S.D. dependent var 9.467683
      - S.E. of regression 1.363844; Sum squared resid 5572.773
      - Akaike info criterion 3.459824; Schwarz criterion 3.467833; Hannan-Quinn criter. 3.462705
      - Log likelihood -5185.736
      - F-statistic 47175.38; Prob(F-statistic) 0.000000
      - Durbin-Watson stat 1.957566

- Annex 3 — OLS combined sample without symmetric links: Average Loss on network size, clustering coefficient, and average minimum path (strongly connected networks note)
  - Combined Sample no symmetric links, OLS: Average Loss on network size (Dependent Variable: LOSS)
    - Date: 10/12/24; Sample: 1 3000; Included observations: 3000
    - Estimated equation: LOSS = C(1) + C(2)*D1*SIZE + C(3)*D2*SIZE + C(4)*D3*SIZE
    - Coefficients:
      - C(1) 25.41284; Std. Error 0.457731; t-Statistic 55.51922; Prob. 0.0000
      - C(2) -0.225445; Std. Error 0.006936; t-Statistic -32.50179; Prob. 0.0000
      - C(3) 0.122138; Std. Error 0.006513; t-Statistic 18.75159; Prob. 0.0000
      - C(4) 0.258827; Std. Error 0.007557; t-Statistic 34.25095; Prob. 0.0000
    - Goodness of fit and diagnostics:
      - R-squared 0.957433; Adjusted R-squared 0.957391
      - Mean dependent var 28.59517; S.D. dependent var 13.51284
      - S.E. of regression 2.789326; Sum squared resid 23309.90
      - Akaike info criterion 4.890810; Schwarz criterion 4.898818; Hannan-Quinn criter. 4.893690
      - Log likelihood -7332.215
      - F-statistic 22462.51; Prob(F-statistic) 0.000000
      - Durbin-Watson stat 1.428259
  - Combined Sample no symmetric links, OLS: Average Loss on Clustering coefficient (Dependent Variable: LOSS)
    - Date: 10/12/24; Sample: 1 3000; Included observations: 3000
    - Estimated equation: LOSS = C(4) + C(1)*D1*CLUST + C(2)*D2*CLUST + C(3)*D3*CLUST
    - Coefficients:
      - C(4) 26.13648; Std. Error 0.434937; t-Statistic 60.09252; Prob. 0.0000
      - C(1) -49.41805; Std. Error 1.383307; t-Statistic -35.72458; Prob. 0.0000
      - C(2) 23.41332; Std. Error 1.297799; t-Statistic 18.04079; Prob. 0.0000
      - C(3) 51.93344; Std. Error 1.514129; t-Statistic 34.29922; Prob. 0.0000
    - Goodness of fit and diagnostics:
      - R-squared 0.954809; Adjusted R-squared 0.954764
      - Mean dependent var 28.59517; S.D. dependent var 13.51284
      - S.E. of regression 2.874028; Sum squared resid 24747.07
      - Akaike info criterion 4.950638; Schwarz criterion 4.958647; Hannan-Quinn criter. 4.953519
      - Log likelihood -7421.958
      - F-statistic 21100.02; Prob(F-statistic) 0.000000
      - Durbin-Watson stat 1.469824
  - Combined Sample no symmetric links, OLS: Average Loss on Average Minimum Path (Dependent Variable: LOSS)
    - Date: 10/12/24; Sample: 1 2075; Included observations: 2075
    - Note: "As noted above, the sample in this case is smaller as it is run only on those networks that are strongly connected."
    - Estimated equation: LOSS = C(1) + C(2)*D1*PATH + C(3)*D2*PATH + C(4)*D3*PATH
    - Coefficients:
      - C(1) 22.18174; Std. Error 0.401420; t-Statistic 55.25821; Prob. 0.0000
      - C(2) -5.977228; Std. Error 0.205676; t-Statistic -29.06141; Prob. 0.0000
      - C(3) 6.834134; Std. Error 0.216716; t-Statistic 31.53498; Prob. 0.0000
      - C(4) 8.556039; Std. Error 0.180972; t-Statistic 47.27819; Prob. 0.0000
    - Goodness of fit and diagnostics:
      - R-squared 0.980187; Adjusted R-squared 0.980158
      - Mean dependent var 29.79590; S.D. dependent var 13.40085
      - S.E. of regression 1.887668; Sum squared resid 7379.573
      - Akaike info criterion 4.110487; Schwarz criterion 4.121355; Hannan-Quinn criter. 4.114470
      - Log likelihood -4260.630
      - F-statistic 34151.51; Prob(F-statistic) 0.000000
      - Durbin-Watson stat 1.216664

*Source: wpiea2025041-print-pdf — Annex Tables and selected excerpts.*

### Annex Tables ...........................................................................................................

### Annex Tables

### Annex 1. Laissez Faire case, OLS
- Title: Laissez Faire case, OLS: Average loss on Network size, Average cluster coefficient, and Average Shortest Path
- Page: 37

### Annex 2. OLS on combined sample
- Title: OLS on combined sample
- Page: 38

### Annex 3. OLS Combined sample without symmetric links
- Title: OLS Combined sample without symmetric links
- Page: 40

*Source: wpiea2025041-print-pdf - Annex Tables (pages 37–40).*

### References .............................................................................................................

### wpiea2025041-print-pdf - References .............................................................................................................

### Executive summary
- Two bank resolution policies compared via simulations using an Abelian sand pile (chip-firing) metaphor:
  - laissez faire: supervisors do not intervene; shareholders absorb failing banks.
  - managed (supervisors’ directed mergers): supervisors direct the strongest institutions (“white knights”) to absorb failing banks via P&A-style operations (each takeover equals ¼ of the failing bank’s balance sheet and reduces the white knight’s capital by 1 unit).
- Main high-level findings:
  - Crises (welfare-loss events) occur less frequently under the managed strategy but are, on average, more costly per event.
  - Over a medium/long term simulation horizon, total social welfare loss is typically lower in the managed case than in laissez faire.
  - The optimal network structure (minimizing average welfare loss per failing institution or frequency of failing events) depends on the chosen resolution policy and supervisors’ objectives.

### Model, assumptions, and scenarios
- Network representation:
  - Directed graph of banks as nodes; links represent shareholder (ownership) relations.
  - Number of nodes in primary simulations: 15.
  - Directed graph built from an asymmetric square matrix (15 x 15) with zeros on the diagonal.
  - Simulations also analyze all possible networks of size 4 (2^12 = 4096 possibilities).
- Capital, thresholds, and shocks:
  - Initial random capital endowment per bank: min of 8 and max of 12 capital units.
  - Prudential minimum capital required: 8 units.
  - Failure threshold: banks fail when accumulated losses exceed 4 (i.e., go below the 8 minimum).
  - Each unexpected NPL shock reduces a bank’s capital by 1 unit; each NPL equals one unit.
- Shareholder loss allocation rules:
  - When a controlled bank fails, each shareholder is allocated ¼ of the failing bank balance sheet and must absorb 1 unit of capital loss.
  - If there are more than 4 shareholders, supervisors randomly select which shareholders absorb losses.
  - When shareholders are less than 4 (or insufficient capital), the unabsorbed portion is liquidated generating a social welfare loss (a “crisis”).
- Two primary simulation scenarios:
  - Laissez faire: shareholders absorb failing banks as above; no supervisor-directed reallocation beyond the shareholder mechanism.
  - Managed: supervisors direct losses to the strongest nodes (those with highest remaining capital) to act as white knights; each P&A reduces a white knight’s capital by 1 unit and white knights initially must have capital at or above 9 to be eligible.
- Additional scenario:
  - Cynical supervisors: supervisors manage failures by directing losses to strong banks but their objective in network optimization is to maximize time between crises (minimize frequency), not total welfare loss.

### Simulation design and parameters
- Typical simulation horizons and repetitions:
  - Primary runs: 300 iterations (interpreted as “days”).
  - For 4-node exhaustive analysis: each network exposed to the same set of 100 shocks.
  - Genetic algorithm (GA) optimization on 15-node networks: population size 26 individuals per generation, 30 generations for the main GA runs (26 x 30 = 780 networks evaluated per experiment).
  - Extended robustness runs: three sets of 1000 simulations for the three bank resolution policies; each of the 1000 simulations runs over 10 generations.
- Network feature measures used in analysis:
  - Size: total number of links/relationships across institutions.
  - Average clustering coefficient: tendency of nodes to cluster (probability banks have common shareholders).
  - Average minimum path: average number of steps to link any two nodes (measure of interconnectedness / “completeness”).

### Key simulation outcomes and quantitative results
- Managed vs. laissez faire (representative observations):
  - Frequency: crises occur less frequently under managed allocation.
  - Severity: average welfare loss per crisis is larger under managed allocation because failures of white knights trigger broader domino effects.
  - Total welfare loss over 300 “days”:
    - Managed total loss typically lower than laissez faire total loss (Figure 6 shows “tot Loss man” lower than laissez faire over full period).
- 4-node exhaustive experiments:
  - Laissez-faire champion: one configuration with average loss slightly over 3.5 units per crisis; crisis occurs on average every 2.37 “days”; total loss over 100 days = 147.68.
  - Managed champions: 64 configurations with minimum average loss of 7.1 units per crisis; crisis occurs on average every 4.84 “days”; total loss over 100 days = 146.69.
  - Conclusion: managed regime yields more networks that achieve the minimum total loss, increasing probability of achieving a best outcome.
- Extended GA and large-sample experiments (1000 simulations each):
  - Total loss distribution (Figure 11):
    - Managed case concentrated: 947 cases out of 1000 concentrated at outcome 2046 (indicating GA found similar optimal configurations repeatedly).
    - Laissez faire average total loss ≈ 2055.5.
    - Cynical supervisors average total loss ≈ 2063 (more volatile; best-case occasionally comparable to managed, worst-case often worse than laissez faire).
- Effect of excluding symmetric links (no direct cross-holdings):
  - Managed case: total losses remain lower than laissez faire; minimum Total Loss for managed = 137 and for laissez faire = 141 (reported in text).
  - Removing symmetric links tends to:
    - Increase frequency of crises under laissez faire (shorter intervals).
    - Decrease frequency of crises under managed rule, especially for cynical supervisors (longer intervals).
    - Not materially change total welfare loss in many cases (total capital in system unchanged), but changes crisis frequency and average crisis severity.
- Strongly connected vs not-strongly connected networks (Table 1):
  - Average Size, Average Cluster, Average Total Loss:
    - Laissez Faire: not-connected Ave SIZE 62.74 vs connected 68.12; Ave CLUSTER 0.30 vs 0.30; Ave TOT LOSS 2055.25 vs 2055.34.
    - Managed: not-connected Ave SIZE 64.54 vs connected 73.86; Ave CLUSTER 0.31 vs 0.31; Ave TOT LOSS 2046.00 vs 2046.02.
    - Cynical Supervisors: not-connected Ave SIZE 60.22 vs connected 61.54; Ave CLUSTER 0.29 vs 0.29; Ave TOT LOSS 2063.89 vs 2064.47.
  - Conclusion: presence of isolated subnetworks alters size and clustering but does not substantially change average total welfare loss across regimes.

### Network optimization and structural insights
- Rugged performance landscape:
  - For 4-node networks, the landscape of average welfare loss is highly non-linear and rugged; small changes in links can produce large shifts in performance (Figure 9).
  - For larger networks (15 nodes), GA used to search for better-performing topologies; 780 examined networks is a small subset of all 2^210 possible structures.
- Which structural features matter (pooled regressions and experiments):
  - No single universal architecture minimizes losses across all regimes.
  - Laissez faire: larger number of shareholding links (network size) helps reduce average loss under laissez faire.
  - Managed regime: regulators minimizing total welfare losses benefit from networks that are:
    - Larger in total number of shareholding relationships,
    - More closely connected (lower average minimum path),
    - More clustered (higher clustering coefficient).
  - Cynical supervisors (minimizing time between crises): networks that minimize frequency of crises often coincide with those that minimize total welfare losses across regimes, but cynical supervisors’ optimization yields more volatile outcomes and frequently worse total loss.
- Symmetric links:
  - Disallowing direct reciprocal cross-holdings (A→B and B→A) preserves many results and can be beneficial under active supervisory strategies by increasing time between crises without automatically increasing total welfare loss.

### Policy implications and recommendations (from the simulations)
- Managed (supervisor-directed merger/P&A) strategies:
  - Tend to reduce total social welfare loss over time despite increasing the average cost per crisis event; simulations support the view that directing mergers is, on net, welfare-improving relative to laissez faire in the modeled environment.
  - May create larger systemic institutions whose failure produces larger domino effects; supervisors should account for systemic buildup when designing resolution frameworks.
- Regulatory treatment of cross-holdings:
  - Prohibiting or netting reciprocal cross-holdings (symmetric links) is generally beneficial under an active supervisory regime: it increases the period between crises and helps supervisors identify stronger white knights more reliably.
- Supervisory objective design:
  - If supervisors optimize for minimizing frequency of crises (politically motivated or “cynical” objective), the resulting networks may produce longer time between crises but often higher total welfare losses and more volatile outcomes; supervisors’ objective functions matter for desirable network architecture.
- Monitoring and macroprudential design:
  - Regulators should consider how resolution practices interact with network topology; possible regulatory actions (e.g., restrictions on cross-holdings, disclosure of ownership networks, capital requirements calibrated to network position) could influence systemic resilience.
- Suggested extensions (simulation/modeling):
  - Use weighted directed graphs to capture magnitude of ownership/control relations.
  - Disallow triangular cross-holdings (A→B, B→C, C→A) to more strictly prevent complex cross-ownership structures.
  - Experiment with higher capital thresholds than the maintained 8 to evaluate nonlinear effects on frequency and total welfare loss.
  - Explore recapitilization and entry dynamics (model currently assumes no new capital creation).

### Key figures and metrics referenced
- Thresholds and capitals:
  - Minimum required capital: 8 units.
  - Failure threshold (excess losses leading to failure): 4 units of accumulated losses.
  - Initial capital per bank: min of 8 and max of 12 units.
  - White knight eligibility initial capital threshold: at or above 9 units.
- Network and simulation sizes:
  - Primary directed graph dimension used in GA and main experiments: 15 nodes.
  - Exhaustive 4-node network space size: 2^12 = 4096 possible directed graphs.
  - Primary simulation length: 300 iterations (“days”).
  - GA parameters: generations = 30; individuals per generation = 26 (total tested networks per run = 780).
  - Robustness experiments: three sets of 1000 simulations each; each simulation runs over 10 generations.
- Representative totals and averages:
  - 4-node laissez-faire champion: average loss ≈ 3.5 units per crisis; average crisis interval 2.37 “days”; total loss over 100 days = 147.68.
  - 4-node managed champions: minimum average loss = 7.1 units per crisis; average crisis interval 4.84 “days”; total loss over 100 days = 146.69.
  - Extended GA aggregate outcomes: managed mode concentration at total loss = 2046 for 947 out of 1000 trials; laissez faire average total loss ≈ 2055.5; cynical supervisors average total loss ≈ 2063.

*International Monetary Fund — wpiea2025041-print-pdf (excerpts provided in source content).*

### Annex Tables

### Annex Tables

### Annex 1 — Laissez Faire, Managed, and Cynical supervisors cases: OLS results for Average loss on Network size, Average cluster coefficient, and Average Shortest Path

- Managed case (Dependent Variable: LOSS_MAN; Method: Least Squares)
  - Sample: 1 1000; Included observations: 1000
  - Estimated equation: LOSS_MAN = C(1) + C(2)*SIZE_MAN + C(3)*CLUST_MAN + C(4)*PATH_MAN
  - Coefficients:
    - C(1) -3.562400; Std. Error 11.28470; t-Statistic -0.315684; Prob. 0.7523
    - C(2) 0.025426; Std. Error 0.039429; t-Statistic 0.644853; Prob. 0.5192
    - C(3) 1.705608; Std. Error 3.944882; t-Statistic 0.432360; Prob. 0.6656
    - C(4) 18.26081; Std. Error 5.315314; t-Statistic 3.435510; Prob. 0.0006
  - Goodness of fit and diagnostics:
    - R-squared 0.067813; Adjusted R-squared 0.065005
    - Mean dependent var 27.52884; S.D. dependent var 2.165686
    - S.E. of regression 2.094114; Sum squared resid 4367.770
    - Akaike info criterion 4.320130; Schwarz criterion 4.339761; Hannan-Quinn criter. 4.327591
    - Log likelihood -2156.065
    - F-statistic 24.15164; Prob(F-statistic) 0.000000
    - Durbin-Watson stat 1.976272

- Laissez Faire case (Dependent Variable: LOSS_LF; Method: Least Squares)
  - Date: 08/28/24; Sample: 1 1000; Included observations: 1000
  - Estimated equation: LOSS_LF = C(1) + C(2)*SIZE_LF + C(3)*CLUST_LF + C(4)*PATH_LF
  - Coefficients (presented as in source):
    - C(1) 23.6977; Std. Error 2.625495; t-Statistic 9.025997; Prob. 0.0000
    - C(2) -0.022957; Std. Error 0.008981; t-Statistic -2.556009; Prob. 0.0107
    - C(3) -2.472345; Std. Error 0.824850; t-Statistic -2.997329; Prob. 0.0028
    - C(4) -5.659769; Std. Error 1.229812; t-Statistic -4.602140; Prob. 0.0000
  - Goodness of fit and diagnostics:
    - R-squared 0.035325; Adjusted R-squared 0.032419
    - Mean dependent var 11.49479; S.D. dependent var 0.439353
    - S.E. of regression 0.432173; Sum squared resid 186.0260
    - Akaike info criterion 1.164008; Schwarz criterion 1.183639; Hannan-Quinn criter. 1.171469
    - Log likelihood -578.0041
    - F-statistic 12.15726; Prob(F-statistic) 0.000000
    - Durbin-Watson stat 2.095158

- Cynical supervisors case (Dependent Variable: LOSS_MH; Method: Least Squares)
  - Sample: 1 1000; Included observations: 1000
  - Estimated equation: LOSS_MH = C(1) + C(2)*SIZE_MH + C(3)*CLUST_MH + C(4)*PATH_MH
  - Coefficients:
    - C(1) 30.15527; Std. Error 3.972491; t-Statistic 7.591022; Prob. 0.0000
    - C(2) -0.004064; Std. Error 0.014649; t-Statistic -0.277464; Prob. 0.7815
    - C(3) 1.114338; Std. Error 1.444334; t-Statistic 0.771524; Prob. 0.4406
    - C(4) 2.239361; Std. Error 1.807025; t-Statistic 1.239253; Prob. 0.2155
  - Goodness of fit and diagnostics:
    - R-squared 0.015869; Adjusted R-squared 0.012904
    - Mean dependent var 33.72440; S.D. dependent var 0.856391
    - S.E. of regression 0.850848; Sum squared resid 721.0458
    - Akaike info criterion 2.518824; Schwarz criterion 2.538455; Hannan-Quinn criter. 2.526286
    - Log likelihood -1255.412
    - F-statistic 5.353304; Prob(F-statistic) 0.001167
    - Durbin-Watson stat 2.077381

### Annex 2 — OLS on combined sample: Average Loss on network size, clustering coefficient, and average minimum path

- 1] Combined Sample, OLS: Average Loss on network size (Dependent Variable: AVELOSS)
  - Date: 10/12/24; Sample: 1 3000; Included observations: 3000
  - Estimated equation: AVELOSS = C(4) + C(1)*D1*SIZE + C(2)*D2*SIZE + C(3)*D3*SIZE
  - Coefficients:
    - C(4) 27.64821; Std. Error 0.401081; t-Statistic 68.93423; Prob. 0.0000
    - C(1) -0.152971; Std. Error 0.003826; t-Statistic -39.98229; Prob. 0.0000
    - C(2) -0.001455; Std. Error 0.003814; t-Statistic -0.381618; Prob. 0.7028
    - C(3) 0.059399; Std. Error 0.003954; t-Statistic 15.02062; Prob. 0.0000
  - Goodness of fit and diagnostics:
    - R-squared 0.973651; Adjusted R-squared 0.973625
    - Mean dependent var 24.24934; S.D. dependent var 9.467683
    - S.E. of regression 1.537591; Sum squared resid 7083.105
    - Akaike info criterion 3.699644; Schwarz criterion 3.707652; Hannan-Quinn criter. 3.702524
    - Log likelihood -5545.466
    - F-statistic 36903.21; Prob(F-statistic) 0.000000
    - Durbin-Watson stat 1.893216

- 2] Combined Sample, OLS: Average Loss on Clustering coefficient (Dependent Variable: AVELOSS)
  - Date: 10/12/24; Sample: 1 3000; Included observations: 3000
  - Estimated equation: AVELOSS = C + C(1)*D1*CLUST + C(2)*D2*CLUST + C(3)*D3*CLUST
  - Coefficients:
    - C 27.05958; Std. Error 0.366463; t-Statistic 73.83979; Prob. 0.0000
    - C(1) -30.96315; Std. Error 0.735540; t-Statistic -42.09579; Prob. 0.0000
    - C(2) 0.864547; Std. Error 0.732593; t-Statistic 1.180120; Prob. 0.2380
    - C(3) 13.54831; Std. Error 0.752456; t-Statistic 18.00545; Prob. 0.0000
  - Goodness of fit and diagnostics:
    - R-squared 0.972738; Adjusted R-squared 0.972711
    - Mean dependent var 24.24934; S.D. dependent var 9.467683
    - S.E. of regression 1.564001; Sum squared resid 7328.515
    - Akaike info criterion 3.733704; Schwarz criterion 3.741713; Hannan-Quinn criter. 3.736585
    - Log likelihood -5596.557
    - F-statistic 35634.00; Prob(F-statistic) 0.000000
    - Durbin-Watson stat 1.884734

- 3] Combined Sample, OLS: Average Loss on Average Minimum Path (Dependent Variable: AVELOSS)
  - Date: 10/12/24; Sample: 1 3000; Included observations: 3000
  - Estimated equation: AVELOSS = C + C(1)*D1*PATH + C(2)*D2*PATH + C(3)*D3*PATH
  - Coefficients:
    - C 18.10670; Std. Error 0.824059; t-Statistic 21.97257; Prob. 0.0000
    - C(1) -4.371290; Std. Error 0.545530; t-Statistic -8.012918; Prob. 0.0000
    - C(2) 6.249760; Std. Error 0.546427; t-Statistic 11.43750; Prob. 0.0000
    - C(3) 10.16341; Std. Error 0.536881; t-Statistic 18.93047; Prob. 0.0000
  - Goodness of fit and diagnostics:
    - R-squared 0.979270; Adjusted R-squared 0.979249
    - Mean dependent var 24.24934; S.D. dependent var 9.467683
    - S.E. of regression 1.363844; Sum squared resid 5572.773
    - Akaike info criterion 3.459824; Schwarz criterion 3.467833; Hannan-Quinn criter. 3.462705
    - Log likelihood -5185.736
    - F-statistic 47175.38; Prob(F-statistic) 0.000000
    - Durbin-Watson stat 1.957566

### Annex 3 — OLS combined sample without symmetric links: Average Loss on network size, clustering coefficient, and average minimum path (strongly connected networks note)

- 1] Combined Sample no symmetric links, OLS: Average Loss on network size (Dependent Variable: LOSS)
  - Date: 10/12/24; Sample: 1 3000; Included observations: 3000
  - Estimated equation: LOSS = C(1) + C(2)*D1*SIZE + C(3)*D2*SIZE + C(4)*D3*SIZE
  - Coefficients:
    - C(1) 25.41284; Std. Error 0.457731; t-Statistic 55.51922; Prob. 0.0000
    - C(2) -0.225445; Std. Error 0.006936; t-Statistic -32.50179; Prob. 0.0000
    - C(3) 0.122138; Std. Error 0.006513; t-Statistic 18.75159; Prob. 0.0000
    - C(4) 0.258827; Std. Error 0.007557; t-Statistic 34.25095; Prob. 0.0000
  - Goodness of fit and diagnostics:
    - R-squared 0.957433; Adjusted R-squared 0.957391
    - Mean dependent var 28.59517; S.D. dependent var 13.51284
    - S.E. of regression 2.789326; Sum squared resid 23309.90
    - Akaike info criterion 4.890810; Schwarz criterion 4.898818; Hannan-Quinn criter. 4.893690
    - Log likelihood -7332.215
    - F-statistic 22462.51; Prob(F-statistic) 0.000000
    - Durbin-Watson stat 1.428259

- 2] Combined Sample no symmetric links, OLS: Average Loss on Clustering coefficient (Dependent Variable: LOSS)
  - Date: 10/12/24; Sample: 1 3000; Included observations: 3000
  - Estimated equation: LOSS = C(4) + C(1)*D1*CLUST + C(2)*D2*CLUST + C(3)*D3*CLUST
  - Coefficients:
    - C(4) 26.13648; Std. Error 0.434937; t-Statistic 60.09252; Prob. 0.0000
    - C(1) -49.41805; Std. Error 1.383307; t-Statistic -35.72458; Prob. 0.0000
    - C(2) 23.41332; Std. Error 1.297799; t-Statistic 18.04079; Prob. 0.0000
    - C(3) 51.93344; Std. Error 1.514129; t-Statistic 34.29922; Prob. 0.0000
  - Goodness of fit and diagnostics:
    - R-squared 0.954809; Adjusted R-squared 0.954764
    - Mean dependent var 28.59517; S.D. dependent var 13.51284
    - S.E. of regression 2.874028; Sum squared resid 24747.07
    - Akaike info criterion 4.950638; Schwarz criterion 4.958647; Hannan-Quinn criter. 4.953519
    - Log likelihood -7421.958
    - F-statistic 21100.02; Prob(F-statistic) 0.000000
    - Durbin-Watson stat 1.469824

- 3] Combined Sample no symmetric links, OLS: Average Loss on Average Minimum Path (Dependent Variable: LOSS)
  - Date: 10/12/24; Sample: 1 2075; Included observations: 2075
  - Note: "As noted above, the sample in this case is smaller as it is run only on those networks that are strongly connected."
  - Estimated equation: LOSS = C(1) + C(2)*D1*PATH + C(3)*D2*PATH + C(4)*D3*PATH
  - Coefficients:
    - C(1) 22.18174; Std. Error 0.401420; t-Statistic 55.25821; Prob. 0.0000
    - C(2) -5.977228; Std. Error 0.205676; t-Statistic -29.06141; Prob. 0.0000
    - C(3) 6.834134; Std. Error 0.216716; t-Statistic 31.53498; Prob. 0.0000
    - C(4) 8.556039; Std. Error 0.180972; t-Statistic 47.27819; Prob. 0.0000
  - Goodness of fit and diagnostics:
    - R-squared 0.980187; Adjusted R-squared 0.980158
    - Mean dependent var 29.79590; S.D. dependent var 13.40085
    - S.E. of regression 1.887668; Sum squared resid 7379.573
    - Akaike info criterion 4.110487; Schwarz criterion 4.121355; Hannan-Quinn criter. 4.114470
    - Log likelihood -4260.630
    - F-statistic 34151.51; Prob(F-statistic) 0.000000
    - Durbin-Watson stat 1.216664

### References (selected from Annex)
- Acemoglu, D., Ozdaglar, A. and A. Tahbaz-Salehi (2015) “Systemic risk and stability in financial networks” Am. Econ. Rev., 105 (2) (2015), pp. 564-608
- Acharya, Viral V. and Tanju Yorulmazer, (2008) Cash-in-the-Market Pricing and Optimal Resolution of Bank Failures, The Review of Financial Studies, Volume 21, Issue 6, November 2008, Pages 2705–2742, https://doi.org/10.1093/rfs/hhm078
- Allen F., and Gale D. (2004) “Competition and financial stability” J. Money Credit Bank., 36 (3), pp. 453-480
- Bak, P.; Tang, C.; Wiesenfeld, K. (1987). "Self-organized criticality: an explanation of 1/ƒ noise", Physical Review Letters, 59 (4): 381–384.
- Elliott, M, Golub, B. and M. O. Jackson. (2014). "Financial Networks and Contagion." American Economic Review, 104 (10): 3115–53.
- Nier E, J. Yang, T Yorulmazer, A. Alentorn (2007) “Network models and financial stability” Journal of Economic Dynamics and Control, Volume 31, Issue 6, Pages 2033-2060, ISSN 0165-1889, https://doi.org/10.1016/j.jedc.2007.01.014.
- White, Phoebe and Yorulmazer, Tanju, (2014) “Bank Resolution Concepts, Tradeoffs, and Changes in Practices” (March 31, 2014). Economic Policy Review.

*IMF WORKING PAPERS Sandcastles and Financial Systems: A Sandpile metaphor — Annex Tables*

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_Source: https://www.imf.org/-/media/files/publications/wp/2025/english/wpiea2025041-print-pdf.pdf_